I have used zelle many times from the browser. It's been a while, so maybe that has changed, though. I never even tried to deposit a check from the browser or an app, so you may be right on that point.
I have 3 different banks (well 2 banks and a credit union.) I can use Zelle in my browser from all 3. I don't even have the app installed for 2 of them.
In my country almost all banks removed their web apps. They existed like 15 years ago, before smartphones became widespread, but nowadays very few banks offer web apps, only mobile apps.