The framers of the Constitution tried to make the limitations on who could run be as minimal as possible since they didn’t want any institutional bad actors messing with the process. This includes being a felon of a state as it was thought a state could convict a candidate without due process for political reasons. If there was something to disqualify a candidate from office, it should be handled through the political process.

Also, it was never really intended for Presidents to be elected. Only two states put the election for President on the ballot initially and South Carolina only let people vote for President after the Civil War.

From that, even when Presidents were directly elected, they were initially nominated by their parties through a closed process. As this was found to be deeply undemocratic, states changed their laws to regulate how candidates were nominated, creating the primary process. Since this was done at the state level, every state did it differently.

So why Trump and Biden? Trump is extremely popular within the Republican Party; he won the majority of primaries and no other Republican candidate could compete with Trump’s popularity. Biden is the sitting President. Traditionally, sitting Presidents only get challenged when they are seen as weak politically. Until the first debate, Biden was seen as strong enough politically that no one wanted to challenge him.