How mature is Avalonia for an universal app? Big 3 desktop plus big 2 mobile?

I can't comment on that specifically, but it works with MVVM extensions toolkit, which is handy for decoupling of event handling and is helpful in complex scenario's.

Most import thing to look for are the components you need imho. You can build themselves, but if you can use something ready made, that helps of course. You would best take look at their gallery to see if you see something similar for your needs.