The answer is simply, "this is how it was designed over 200 years ago, and it's (deliberately) very hard to change it":
- The US Constitution describes (in a fair amount of detail) the process by which the President is elected, and does provide a method for removal: Impeachment.
- Because presidential election and removal is detailed in the US Constitution, Congress can't simply pass laws to change how it works, we would have to change the constitution itself. This is a (deliberately) difficult process requiring super-majority approval of both Congress and US state legislatures: Procedures for Amending the US Constitution.
- Since then, only one proposed amendment dealing with presidential succession has garnered enough support to pass: the 25th Amendment, which was passed in 1967 in the wake of JFK's assassination. This amendment deals mainly with what happens when the president is incapacitated.
As other answers have noted, "high crimes and misdemeanors" as a reason for impeachment is a phrase with broader meaning than just literal crimes:
The convention adopted “high crimes and misdemeanors” with little discussion. Most of the framers knew the phrase well. Since 1386, the English parliament had used “high crimes and misdemeanors” as one of the grounds to impeach officials of the crown. Officials accused of “high crimes and misdemeanors” were accused of offenses as varied as misappropriating government funds, appointing unfit subordinates, not prosecuting cases, not spending money allocated by Parliament, promoting themselves ahead of more deserving candidates, threatening a grand jury, disobeying an order from Parliament, arresting a man to keep him from running for Parliament, losing a ship by neglecting to moor it, helping “suppress petitions to the King to call a Parliament,” granting warrants without cause, and bribery. Some of these charges were crimes. Others were not. The one common denominator in all these accusations was that the official had somehow abused the power of his office and was unfit to serve.
In my opinion, this means that the current Congress, if it so desired, has the full ability to remove a sitting president under the current rules. Why they have not done so yet is political speculation outside the bounds of your original question.
Other follow-up questions come to mind, as well, some of which are dealt with in other answers. (Questions like "why did the founders choose impeachment vs other methods?" or "how difficult would it be to hold a national presidential election in an off year?") In my opinion, the founders thought it was more important for the office of the president to be filled immediately upon vacancy than to wait for a new election.
It is possible that at some time in the future, as we look back upon the legacy of Trump's election and presidency, that there will be sufficient support to propose and pass other Amendments regarding presidential elections or removal or succession. But right now, especially in the midst of this chaos, it is impossible to speculate what those might be.