There are some tricks which work for different websites - for example, for NYT it's enough to manually clear nytimes.com cookies, FT used to work after click from twitter/x and so on. So I guess there is some set of heuristics.

I always assumed so - because Google can index them full text. It used to be the case that you could see those full snapshots in Google as cache - this was before the sites strong armed Google to remove those snapshots from being accessible, then archive.* folk rose to power. You can test this yourself for searching for a unique quote on those sites and still getting hits in Google. But you are right - why could this not be achieved with a plugin then - don't know.

Perhaps not in the same way as described above, but BPC exists.

https://en.wikipedia.org/wiki/Bypass_Paywalls_Clean