but it’s not as bad as if it was run by Trump and co.

The U.S. isn't a functioning democracy though, which is why that's a problem. And just because a nationalized service is controlled democratically doesn't mean it is controlled by a president. There are a lot of different ways to have democracy.

And we no longer live in an era of horse and buggy, so democracy can be far more direct than it has in the past.

In addition, there is already a multitude of positions filled/appointed/approved by the president. The administrator of NASA, the administrator of the EPA, etc. There is nearly 500 federal agencies like this.

So this would not be a problem unique to a nationalized search engine. So the solution is an actual democratic control of these agencies/administrators, not a wanna be dictator.

Another thing to keep in mind, what I'm proposing is something that would only ever work in an actual functioning democracy. So therefore I am not proposing this within the U.S.

Which you didn’t include, is splitting Google up

As I said, I think it is debatable if a search engine is even critical enough to warrant nationalization. I don't think the need is there. And as I (admitted retroactively edited my comment to say), I have previously stated that I'm totally cool with breaking up Google at a bare minimum. The rest of this is just about the hypothetical of nationalization.

Split the advertising from search.

Short of publicly funding private companies, this would just result in a subscription model, which nobody wants. It's either ads, subs, or public subsidization.

This is the surest way to make them cater to us.

It's a half measure. The only real way to make them cater to us (aside from previously mentioned nationalization) is regulation, workplace democracy, and so on.

Even if Google got turned into a small company that only ever does search, they'll still be a business running under capitalism, with all of the profit seeking motives that got us to where we are now.