"even more sites would be destroyed"

Why? If you make a law, that people get a finders fee if they find something of historic value, only if they stop digging after finding it, but get a fine if they do destroy the site, how would that make more destruction?

Harder is it for homeowners, because real compensation there costs real money - but how else would you compensate people for individual loss, for the common benefit (research)?

I'm thinking, more people could go "hunting" that way.

And if they do so in a controlled, legal setting, where they cooperate with proper scientists, wouldn't that be way better?

Not if it compels them to hold their land fallow for an indeterminate months or years, in exchange for a one-time payment (or none at all). I’m all for proper archaeology but I can’t expect those who are trying to use their land to prioritize it unpaid. I don’t expect their government offers enough social support and compensation for the lost time to make it worth the risk and annoyance.