The President Has To Deal Directly With The Consequences On Enacted Legislation
In General
The President's job, beyond partisan legislative priorities, is to make the vast federal bureaucracy function and to carry out the laws on the books.
As the person who has to deal with whatever Congress has wrought, at a minimum a President's administration will almost always want to take an active part in negotiations over any law that the executive branch will have to play a material role in administering.
This is because even slight details and fine print can dramatically change how difficult or easy it is to administer a law.
Debt Ceiling Legislation Is Particularly Problematic
A debt ceiling bill is particularly problematic in this regard.
Even if the House passed debt-ceiling bill did not specifically target the President's recently enacted legislative accomplishments, the President would still have a strong interest in intervening to secure a solution because a debt-ceiling bill, by its very nature, conflicts with legislation from appropriations bills already enacted by Congress in every instance where it matters at all.
When an appropriations bill is passed, Congress has authorized the President to implement that bill by entering into commitments to spend federal funds for particular purposes, and the President, if he is trying to "faithfully execute the law" has begun to implement those bills.
A debt ceiling bill essentially seeks to roll back existing appropriations bills upon which commitments have already been entered into by the President to spend funds in furtherance of carrying out the task of spending funds appropriated by Congress for particular purposes.
While a President is only rarely required to spend all funds that an appropriations bill authorizes the President to spend, any debt ceiling bill places a President in a tough position when trying to run the federal government. So a President of any party has a strong incentive to play an active role in negotiating the terms of, and marshaling lobbying on behalf of the administration regarding, any debt ceiling bill.
There Is Some Hotly Contested Legislation That A President Can Choose To Ignore
There are hotly contested bills about which the President could, if so inclined, take a back seat in the legislative process, because they have little impact on the day to day administration of the federal government.
For example, a reform of the substantive terms of the federal Junk Fax law, or federal laws governing jurisdiction over which state's courts get to decide child custody cases, which are entirely administered by state rather than federal courts, are matters that the President could punt upon, if so inclined.
But bills with minimal implementation concerns are the exception, more than they are the rule.
The President Has The Most Power Of Any Participant In The Legislative Process And With Power Comes Moral Responsibility
President's Care About Policy And The President Is The Most Powerful Legislative Process Participant
Of course, Presidents, in addition to being charged with implementing almost all Congressional enacted law, is also a leader of the President's party who ran on policy positions.
The President's veto power means that no legislation can be passed if it is opposed by the President and either of the two major political parties or by a significant bipartisan minority even if supported by majorities in both parties, which gives the President far more negotiation clout than anyone else in the legislative process. So, of course, Presidents are not going to want to let that power go to waste.
And, any time someone has the power to influence important decisions impacting the fate of the country, a moral responsibility to do so in a wise and proper manner comes with that power.
The Founders Could Have Written A Constitution Where The President Was Less Powerful
If the Founders had wanted a constitution in which the President simply did what he was told, they wouldn't have given him the veto power and would have had a President elected by the U.S. House, and would have positioned the Senate as the source of any veto-type power while denying it the power to initiate all or most kinds of legislation (which is essentially how bicameral parliamentary systems work).
Another common modern solution is to grant almost all power to the legislatively elected Prime Minister with a President serving only the role of a symbolic head of state similar to the modern U.K. monarch. But, in 1789, the notion of a monarch or a monarch-substitute with only symbolic power was merely a pipe dream of political philosophers and wasn't nearly as obvious a solution as it was to later constitution drafters in the era in which the British monarch had ceded all but symbolic power.
Most of the Founders assumed that the balance of power would be more strongly in favor of Congress and less strongly in favor of the President. So, they tried to weaken Congress vis-a-vis the President (while weakening the Presidency relative to the formal but evolving and declining power of the monarch) and got a more powerful Presidency than they bargained for.
There Are Good Reasons That The Founders Failed To Accurately Predict How Their Constitution Would Work In Practice
The Founders can be forgiven for their misreading of how their rules would play out.
The Founders overestimated the strength of Congress relative to the President, in part, because their primary reference point was the U.K. constitutional monarchy at a moment in U.K. history prior to true parliamentary supremacy and before it became a de facto unicameral majority rule system, in which neither the monarch nor the House of Lords had much power.
The decline of the direct powers of the monarchy in Britain to what became mostly a symbolic role took place mostly after 1789:
From 1811 to 1820, George III suffered a severe bout of what is now
believed to be porphyria, an illness rendering him incapable of
ruling. His son, the future George IV, ruled in his stead as Prince
Regent. During the Regency and his own reign, the power of the
monarchy declined, and by the time of his successor, William IV, the
monarch was no longer able to effectively interfere with parliamentary
power. In 1834, William dismissed the Whig Prime Minister, William
Lamb, 2nd Viscount Melbourne, and appointed a Tory, Sir Robert Peel.
In the ensuing elections, however, Peel lost. The king had no choice
but to recall Lord Melbourne. During William IV's reign, the Reform
Act 1832, which reformed parliamentary representation, was passed.
Together with others passed later in the century, the Act led to an
expansion of the electoral franchise and the rise of the House of
Commons as the most important branch of Parliament. The final
transition to a constitutional monarchy was made during the long reign
of William IV's successor, Victoria.
Similarly, the inferior status of the House of Lords was formally institutionalized in the Parliament Act of 1911 and 1949.
There were precious few examples of previous attempts to create anything all that similar to the kind of political structure that they created in 1789.
They also didn't fully foresee the impact of the end of the unifying impact of the revolutionary war, the the impact of having someone other the unifying revolutionary figure of President Washington in the Presidency when trying to see how politics would play out later on.
While they certainly foresaw that there would be factions that any political system would have to manage, they naively overestimated the extent to which deliberation and reason and apolitical good will on the part of politicians would mitigate partisan conflicts between two major parties at a time. It turns out the partisanship is more powerful, and independent reasoned deliberation is less powerful, in the political system that they created (as it evolved over time) than they predicted that it would be.
And, certainly, no one in 1789 had any inkling that their constitution would still be mostly in place in 2023 (234 years later) with 50 states and 336 million people ruled by it in a nation extending from deep into the Pacific Ocean to the Atlantic Ocean's Coast and the Caribbean sea, with a near universal franchise, and an economy primarily rooted in vigorous interstate and international commerce with modern 21st century technology. Today, there are more federal employees alone than there were people in the U.S. in 1789.
By comparison, consider what North America looked like politically, or what democratic politics more generally, looked like at a global level, in 1565 (234 years before the U.S. Constitution was adopted).
A map of the Western Hemisphere in 1562
(Source for map)