I'd simply use "bounty" tags on issues, like some cryptocurrency projects do
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Is it not ok to put the bounty there in writing and send the money without the help of another service?
I guess I would be ok sending the money on my own. But these other services have the nice feature of allowing other users to contribute to the bounty into a single pot. I.e., I can put a bounty of $20, then user b also really wants the feature and will add $5. When the PR is approved, the developer is guaranteed $25 and doesn't have to contact user b to send their $5 or give out their financial info to X number of people.
The first bounty I would create for any project of interest is a bounty to move the bug tracker out of Github so those who boycott Microsoft can at least participate in the QA process.
So having a bounty mechanism inside Github is a bad idea. As a MS boycotter, I would be excluded from contributing bounties via the mechanism you propose.
Well, I'm using GitHub. I don't know what to tell you.