The Pre-Modern Pardon Power
It is worth noting that direct appeals of criminal convictions are relatively recent.
In federal courts prior to the 1890s, the only judicial remedy you had if you were convicted of a crime was a habeas corpus petition which largely tested whether the court had jurisdiction over your case, and whether you were being detained without receiving a speedy trial or after being acquitted.
The innovation of allowing direct appeals of criminal convictions came to state courts at different times, but generally in the same era. Even when states started to authorize them, in many states back then, the sole appellate court was the state supreme court with a purely discretionary docket of criminal appeals (similar to the U.S. Supreme Court docket today).
Prior to the advent of direct appeals of criminal convictions, the pardon power was used much more expansively, and mostly to address convictions that were likely to have been wrongful.
The stakes were also higher in the early pre-modern period when the death penalty was the norm for most serious felony offenses, and not just murders and a handful of other crimes often carrying a great risk of causing a death. With death at stake and few options for a judicial appeal, there was a much greater moral obligation for Governors to avert a travesty of justice by pardoning criminal defendants in doubtful cases.
Also, of course, the mediocre state of forensic science meant that the accuracy of guilt finding that was possible in a criminal jury trial was often much lower than it is today.
In part, this pre-modern regime reflected the fact that society didn't have the economic resources to incarcerate serious felons securing for the many, many years of incarceration that we now think is appropriate for serious felonies. Society also didn't have the economic resources to incarcerate convicted criminals for long periods of time to allow criminal appeals to be litigated as we do now.
Driven by similar economic considerations, harsh corporal punishments short of the death penalty or exile were also common penalties for less serious felonies and serious misdemeanors.
A discretionary pardon, in contrast to a judicial appeal, could provide relief in appropriate cases in the short time available before a sentence of death or corporal punishment was carried out (frequently within days to a couple of months after the criminal defendant was convicted).
FUN FACT: The U.S. Constitution still does not guarantee a right to a direct appeal of a criminal conviction, although habeas corpus relief remains available and addresses more cases than it used to, and if the government legislatively gives you a right to a direct criminal appeal at all, it must meet certain constitutional requirements in doing so.
While this all goes to the pre-modern practice of using the pardon power, you can find what the Founders thought about the pardon power when the U.S. Constitution was being written as a matter of theory, for example, here:
On the afternoon of Wednesday, June 18, 1788, George Mason rose from
his chair on the floor of the Virginia Ratifying Convention deeply
troubled by what he thought of the convention’s failure to
understand—the president of the United States might not always be
someone of sound character and high intelligence. There would rarely,
if ever, he reminded the delegates, be a commander in chief with the
courage and rectitude displayed by George Washington during the War of
Independence. There might even be a president who would try to change
our form of government. The president, argued Mason,
“ought not to have the power of pardoning, because he may frequently
pardon crimes which were advised by himself. It may happen, at some
future day, that he will establish a monarchy, and destroy the
republic. If he has the power of granting pardons before indictment,
or conviction, may he not stop inquiry and prevent detection? The case
of treason ought, at least, to be excepted. This is a weighty
objection with me.”
Some of the most famous men in American history were there that day as
delegates to the Virginia convention. Patrick Henry, afraid that a
national government would destroy the states, was leading the fight to
reject the Constitution. John Marshall, who, as Chief Justice of the
Supreme Court, would do more than anyone to make the Constitution the
foundation for the kind of strong national government Henry feared,
was one of the leaders in the fight to ratify it. But there was no
one—no one in Virginia, nor in the country—with a deeper understanding
of the Constitution and what it meant than James Madison.
Madison understood immediately the force of Mason’s objection, but he
had a response—a response in which he described limitations on
presidential power that, to our great misfortune, have for too long
been forgotten. Was there a danger in giving the president the power
to pardon? “Yes,” replied Madison, but there was a remedy for the
danger in the Constitution as drafted.
“There is one security in this case to which gentlemen may not have
adverted: if the President be connected, in any suspicious manner,
with any person, and there be grounds to believe he will shelter him,
the House of Representatives can impeach him; they can remove him if
found guilty.”
The Modern Pardon Power
In the modern era, a direct appeal of a criminal conviction of right, followed by a discretionary appeal of a criminal conviction to a state supreme court and possibly the U.S. Supreme Court, followed by a state habeas petition and appeals from that, followed by a federal habeas petition and appeals from that are potentially available.
Coinciding with the rise of judicial branch appeals, the number of pardons of people who have not fully served their criminal offense sentences has grown vanishingly small at both the state and federal level (except for policy changes like mass pardons of draft dodging offenders following the Vietnam War, or several mass pardons of people guilty of marijuana possession), although the decline in the use of the pardon power lagged behind the growth of access to appeals.

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Even when the pardon power is used at all these days, the vast majority of pardons are of people who have fully served their criminal sentence and shown evidence of being reformed after being released, to allow those people to be free of the collateral effects of a criminal conviction like not being able to vote or own a gun or enter a regulated occupation.
There are a trickle of pardons or commutations of people who haven't finished serving their sentences in cases where the original sentences seemed harsh or guilt is cast into doubt and the courts are unresponsive. But they are now exceedingly rare.
Also, while the federal government vests the pardon power in the President, and many states vest comparable pardon power in Governors, this isn't universal.
Many U.S. states have a more limited pardon power (e.g. only allowing pardons of people once they are convicted of crimes), and a minority of U.S. states vest the pardon power entirely in a board of pardons and paroles, or requires a Governor's pardons to involve cooperation of some sort from a board of pardons and paroles.
A 50 state survey of state pardon systems can be found here. The District of Columbia, the federal system, and three U.S. states are basically completely unstructured. In 19 states, the Governor may pardon unilaterally, but has a board of pardons to provide optional advice to the Governor. In 8 states, consultation with the pardon board is mandatory but the Governor isn't bound by its advice. In 4 states, the Governor is on a pardon board and the decision is made by the Board collectively. In 10 states, the Governor can only consider pardons that the pardon board approves. In 6 states pardons are made by a pardon board without involvement from the Governor. And, in addition to these general rules some states have some exceptions or qualifications of the general rules (per the linked survey of pardon policies above):
In states which usually don't involve the Governor with pardons at all:
In Alabama and South Carolina, the governor remains responsible for
clemency in capital cases, and in Idaho, the governor must approve the
board’s decision to pardon certain serious crimes.
In states with "gatekeeper boards":
In Rhode Island, the senate must advise and consent to every pardon.
In South Dakota, the governor has constitutional authority to pardon
without consultation with the board, but sealing is unavailable to a
grantee if the statutory procedure requiring board approval is not
followed. The result is that in recent years all pardons have been
granted after board approval.
California's Governor generally has unilateral pardoning power with a board that the Governor may, but is not required to consult it, but there is an exception to that rule:
In California, the governor is required to consult with the parole
board and seek approval of the state supreme court in recidivist cases
only.

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The variety of U.S. state level criminal justice rules matters quite a lot because something on the order of 98% of criminal convictions occur in state courts which the President can't pardon, while only about 2% of criminal convictions occur in federal courts and can be pardoned by the President.
Very little serious "blue collar crime" that occurs outside of Indian Reservations is handled in federal court. Bank robbery and interstate kidnappings are some of the main exceptions to this general rule.
OBSERVATION: The dramatically reduced use of the pardon power also coincides with another notable political development. In the premodern era after the U.S. Constitution was adopted, the U.S. House routinely resolved disputed Congressional elections itself (almost always on party-line votes). But around the same time that more appeals were provided for criminal defendants, there was a fairly dramatic shift towards having courts resolved disputed elections, and Congressional resolutions of disputed elections became vanishingly rare.
Indeterminate v. Determinate Sentencing Considered
A final twist that is worth recognizing is that while the federal criminal justice system in the U.S. and some states have mostly determinate sentencing (which means that the length of the prison term is mostly decided when you are sentenced by a judge for a crime), many U.S. states mostly have an indeterminate sentencing system for serious crimes.
In an indeterminate sentencing system, the sentencing judge sets a minimum sentence and a maximum sentence that are quite far apart from each other, and a parole board in the prison system decides at what time between the minimum and maximum date set by the judge you are actually released from prison with your behavior while incarcerated playing a significant role in that decision.
In a sense, in an indeterminate sentencing system, commutation (i.e. reduction) of prison sentences becomes the norm and serving a maximum prison term becomes the exception.