I should clarify that I meant "webmail" in general, and meant to answer the spirit of the broader question about why people prefer thick clients over web clients. Most web clients aren't great offline, but a "YMMV" does seem necessary here.
With that said, while I don't know much about the particulars of the Proton web client and how it handles offline access, I'd still prefer a client that treats all mail the same way, regardless of provider, and that I can manage locally/migrate to a new computer/back up to my own NAS, etc.