There was also the dot-com boom, which of course wouldn't have happened without Al Gore inventing the Internet.
Biden had to deal the tail end of a pandemic and the financial fallout of that.
Not saying both couldn't have done much better on the budget but, if you compare them to Bush and Trump, they honestly do look quite competent.
Structurally, the budget has gotten worse every year since the 1990s. In large part for reasons not attributable to presidents. For example the number of old people (as a % of the population) in the US has doubled since the '90s, thus ballooning medicare and social security spend. Without cutting eligibility to those programs, increased spend was always going to be an inevitability due to seismic underlying demographic trends. Second, the persistent deficit has led to now $2T in annual interest expense. Again, not something an administration can simply "reform" or back out of; that is now structural inherited spend.
Then Republicans took Congress in 1994 with their "Contract With America". They passed a budget that was much closer to balanced, whether Clinton wanted it or not. Eventually, as you say, that led to a surplus in 2000. (That was the first surplus since Andrew Jackson, by the way.) But Clinton by himself wasn't going to do it. He was going to play pretend. It was the Republican Congress that forced him to.
Now, if you think I'm advocating for the Republicans here, I note that Republicans don't believe in anything like a balanced budget when the president is their guy. I'm just saying that, if you're going to point to Clinton, state the record accurately.
(One detail I don't remember: Why did Clinton's 1993 deficit plan get no Republican votes? Was it because they did not want to cut the deficit? Was it because they just wanted to deny Clinton a victory? Or was it because they wanted to actually do something about the deficit, and Clinton's plan was more pretend than real?)