[ben_w]

There is no other (let alone better) option — those cover all possibilities.

If you don't check papers (or an equivalent), you automatically don't control immigrants at all once they're in your borders. To control them requires everyone has an ID, because otherwise some immigrant who doesn't want to have ID for whatever reason can just say "of course I don't have papers, only immigrants have papers and I'm not an immigrant".

The entire point of papers is to determine if someone is or isn't in some category (including but not limited to "immigrant").

What you can do is automate ID checks. So long as you don't mind your ID being tested at random, and the way that such systems have a tendency to randomly fail and deport people who were actually allowed to remain.