Why are we okay asking soldiers to die, but asking the President to risk prosecution and be subject to the law is "too much"?

Because we need governments and can't afford to make it so that anyone who ever wins elections spends the rest of their life fighting off lawsuits. Wars happen, and governments exist, people forget that laws are used as much to impede good governance as they are to enforce good governance.

What would be the consequences if we did allow Presidents to be prosecuted and some of them did spend the rest of the life fighting prosecutions?

It seems to me the consequence would be that we would have Presidents who act carefully to not break the law. Are there any other consequences?

I've heard it said that "democracies are where the people in power lose". We've all seen "democracies" where the party in power never loses; those aren't real democracies. Likewise, I would say nations that are truly governed by law are where those in power face legal prosecution.

(Also, what's so bad about prosecution? If a person is innocent, the legal system will find them to be innocent, right?... Right?)

The consequences would be Lawfare.

And what's the consequence of that? Presidents who are extra careful to obey the law because they know they will be highly scrutinized?

Or, to put it another way:

There are good and competent people who say "I'm willing to be President and be subject to the law, I'm willing to do both". Why can't be pick a President from that group?

Because doing so gives the President tons of incentives for bad behavior. It's a fun case where "risk of prosecution" would result in tons more questionable behavior then less.