This is for iPhones only. Do I assume correctly that the issue is solved for Android, instead?

I don't know the specific ones, but there are android apps that replace Google Messages (dark pattern warning: enabling this makes your previous messages inaccessible as Google Messages won't let you read them unless it's your default SMS app) as well as apps that can send and receive in parallel to Google Messages. I'm not sure if there are also apps that can read your message history out of Google Messages.

I've been using SMS Backup & Restore (from SyncTech) for years - it reads full history out of Google Messages any time I've used it.

Also curious. I keep messages.google.com in a pinned tab but the number of times I have to go find my phone and re-authenticate is too damned high.

On Android, you can use KDE Connect on KDE and GSConnect on GNOME. It has a Messaging app that syncs with your phone's SMS.