Haven't seen any of those on Android so far, so I don't see why they would appear for iOS. Hell, I've barely even seen websites do that on desktop (Microsoft Teams being a notable problematic web app here) and I'm on Linux+Firefox. The only real exceptions so far have been applications that choose not to support Firefox because of its abysmal market share. It's often easy to find working alternatives for those, though.

The only reason devs would show a "download chrome to use this" prompt would be if Safari were behind in support for the necessary features or if Safari's bugs are too much of a hassle to support (i.e. lack of push notifications, lack of web FS/BT/serial if the web app needs that, or a lack of provisioned browser storage that the mega.nz website relies on).

What I expect to happen, and what the recent expanse of the webkit team suggests is already happening, is that Safari will get better once it gets competitive pressure from Chrome and other browsers.