When was it looked into? The Zulip mobile app was rewritten in Flutter recently, that version was in beta for several months and was finally made the default Zulip app about a month ago. I haven't used Mattermost so can't compare, but the Flutter Zulip is much more responsive and nice than the previous Zulip app.
Our Flutter experience over the last few months since launch has been very positive. Most importantly, development velocity is much faster than it was on React Native.
Looks to be some sort of subscription licensed framework, and lacks desktop support. Why should I move off an open source platform onto a hosted solution? Especially in the context of OP’s situation.
I believe Flutter was chosen because of somewhat easy way of keeping common codebase for both iOS and Android clients. Not trivial, and at least it renders natively :V
Flutter "compiles to native," but the UI is just a giant canvas they paint themselves. React Native uses real native views, so you get actual platform widgets, accessibility, and OS-level optimizations instead of shipping your own game engine.
Also, Google has a habit of hyping projects then quietly killing them (I sadly took the Polymer ride).
I liked Zulip a lot until that Flutter rewrite. Maybe it's more accessible now but the new look is not for me. I believe the app navigation is largely unchanged, and still doesn't quite feel right. I love the topic-based model nonetheless.