Would it help if the President were required to cut party ties?

While it sounds like a great idea on the surface, it would require a total revamp of our entire political system. As discussed in a previous thread, it takes an enormous amount of money to be a viable candidate, let alone win a presidential election and the vast majority of those dollars come through the political party...this is the root of the evil of our political system. Quid pro quo. The 'party' got you elected, therefore you owe the party. Right or wrong is another debate entirely, but it is what it is.
Also, how would such a policy be enforced? There would be continuous allegations of the sitting president 'favoring' the party of their political roots, whether substantiated or not, which would lead to increasing distractions from the major issues...it could even be used as a 'red herring' to distract from issues.
Also, any candidate is affiliated with a certain party due to their own personal beliefs and values. Are you saying that they should put their own values and beliefs aside? Easier said than done.
Again, while it sounds good; and would be a noble thing to do, I don't see it as a possibility.

As many have already said, this is a good idea in theory, but the truth is that, regardless of whether the President renounces his/her political affiliation or not, he/she will always be identified with the party that put him/her in office and will be his/her prime supporter for reelection. (Yes, I know -- a lot of he/she and him/her, but it is a reflection of today's reality).

What we really need is a President to work with everyone, regardless of political party, to get the job done. Although we have had a few do that, they have been very few and far between. Our present system is pretty dysfunctional. Selfless, non-partisan service by politicians is a rare commodity.

Have you ever heard a phrase similar to "You can take the man out of the _____, but you can't take the _____ out of the man."? This is true with political parties. Candidates joined those parties because the are aligned with their personal belief system. Forcing renouncement of a political party won't remove bias or beliefs simply labels.