It feels like every pick of this administration is just someone who has a motivation for corruption.
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No; the problem with Trump is the specific types of laws he broke. He broke laws around honesty, fair dealing, obstruction of justice, and, of course, the integrity of elections and our democratic process. He is, very plainly, opposed to democracy and the rule of law, preferring to replace it with cronyism and nepotism.
That's why he's bad for the country, not simply "because he's a convicted criminal."
But there's only one POTUS. There are multiple people out there who could do a decent job of being POTUS. Many of them are not convicted felons. We wouldn't lose much if we filtered out the entire convicted-felon category from this particular job.
He gets the police to arrest this opposition candidate, let's say for marijuana possession with the intent to sell (a felony), and with procedural chicanery ensures that the court trying the case is run by a Trump-appointed judge.
The opposition candidate is convicted of this charge.
Under your suggestion, they would then become ineligible to be elected.
It gets even worse if the corrupt President has a compliant Congress (which it seemed like he did for a little while, but that's less sure now). If he can ram through a new law making "criticizing the sitting President" a felony, then basically anyone who would oppose him and his regime would clearly be guilty.
In general, the sitting government decides what is a crime and what is not. If you make a law that says that those convicted of crimes cannot run for public office—either "any public office" or "only this specific public office"—then the sitting government, if it is seeking to act in its own interests rather than those of the people, has a perverse incentive to preferentially criminalize things that those who disagree with them are more likely to do, and to encourage (or coerce) bias in policing and trials to ensure conviction.