I would definitely say the web client is less polished currently, and we are holding back there until some other pieces are in place. But I will also say it has a kind of document editing direction and we found that animations felt sluggish and more in-the-way in that context, so they would be more subtle.
The glyph is interesting, had you joined TestFlights in the past or just recently for this syncing? We will look into it here. It actually is associated with live list sharing but this is supposed to be internal only 
(It’s not coming soon, much further out if it happens!) Should make sure it’s totally shut down ahead of shipping initial sync.
Let me look into the adding items question. I like the idea of just typing and it starts entering an item at the top (if nothing else is in focus) but will have to do some testing if this is a viable direction.
Oh and yeah, the saving web apps feature is nice! I’ve been using it that way too, should definitely highlight it in the upcoming beta thread.