It is true that browsers do much more computation than "dumb" terminals, but there are still non-trivial parallels. Terminals do contain a processor and memory in order to handle settings menus, handle keyboard input and convert incoming sequences into a character array that is then displayed on the screen. A terminal is mostly useless without something attached to the other side, but not _completely_ useless. You can browse the menus, enable local echo, and use device as something like a scratchpad. I once drew up a schematic as ascii art this way. The contents are ephemeral and you have to take a photo of the screen or something in order to retain the data.
Web browsers aren't quite that useless with no internet connection, some sites do offer offline capabilities (for example gmail). but even then, the vast majority of offline experiences exist to tide the user over until network can be re-established, instead of truly offering something useful to do locally. Probably the only mainstream counter-examples would be games.
Google's own Gemma models are runnable locally on a Pixel 9 Max so some lev of AI is replicatable client side. As far as Gmail running locally, it wouldn't be impossible for Gmail to be locally hosted and hit a local cache which syncs with a server only periodically over IMAP/JMAP/whatever if Google actually wanted to do it.
The gain, as far as local AI goes for Google, is that, at Google scale, the CPU/GPU time to run even a small model like Gemma will add up across Gmail's millions of users. If clients have the hardware for it (which Pixel 9's have) it means Gmail's servers aren't burning CPU/GPU time on it.
As far as how Gmail's existing offline mode works, I don't know.
To make the discussion easier, we can look at the state of GMail before widespread AI was a thing.
(I have to use some weaselwording here, because GMail had decent spam detection since basically forever, and whether you call that AI or not depends on where we have shifted the goalposts at the moment.)
Heh that brings to mind that old chestnut, something like "We call it AI until it works, then we call it machine learning." Google's running the servers tho, so the assumption is the servers would be running the spam filter against it before the user. In my mind, anyway. So it's fair to point out that Google also has the compute resources to not have to care. To the hypothetical though, could Google make Gmail a local-first? I say yes. Will they? I doubt it.
