Apparently so.
Being not a security person or authentication-knower, couldn't they use... (OAuth? or another of these things I've vaguely heard of?)... to let you sign in on one server using your account from another? Even if I had to do this for 20 different servers, *once*, it'd still be a great improvement.
Obviously since they haven't, I'd guess they probably couldn't, but.
There is a github issue about it: https://github.com/mastodon/mastodon/issues/19652 and apparently the behavior has been changed as of Mastodon 4.0, which is presumably what @kha had in mind in their earlier reply to me.
Update: I discovered that if you *left-click* on a thing in the timeline, it opens a view of it that's still on the same server, so interaction works. It's only middle click to open a new tab that goes to the originating server. The internet has thoroughly trained me out of ever left clicking anything on a website with infinite scroll.