Foreigners get honorary knighthoods: Bill Gates, Steven Spielberg, Dwight Eisenhower etc. Apparently if you take up British Citizenship it automatically converts.

That seems to be a US rule [1], not a condition imposed by the knighting jurisdiction. Maybe some other countries have similar no-titles rules, but I couldn't find any (might be searching for the wrong terms).

[1] https://en.wikipedia.org/wiki/Foreign_Emoluments_Clause

Well, a knight is a servant to the monarch, thus subject of her.

Knighthood is not just a title, it also means you are servant, subject to the monarch. My understanding is you have to become citizen for this.

And while Britain allows dual citizenship, most countries do not. So in most cases you need to rescind your other citizenship in order to be come an actual knight.

Not quite true. If you wished to be addressed by the title of 'Sir' or 'Dame,' you would have to be a British citizen, but a non citizen can add letters after their name, e.g. Bob Geldof, KBE, remains a citizen of the Republic of Ireland.

I believe Commonwealth citizens (at least, those from countries that share a monarch with the UK) are entitled to use 'Sir' or 'Dame' as well: though Canada, for one, seems to take a dim view of its citizens accepting British honours.