Most paywalls just allow search engines to read their content just fine. Because they do want discoverability, they want their cake and eat it.
There's a few publications that don't even do that though and archive.is is very good at bypassing them so I do imagine they use logins for those, but for the masses of sites it's not currently necessary.
Then why hasn't anyone built a client-side browser addon that impersonates a suitable search engine?
You can't impersonate Google. Sites check the source IP and they don't overlap with Google Cloud.