I have worked on Dart and Flutter for a long time. One question I was recently asked was whether Flutter is a suitable technology when it comes to making modern web apps.
The answer is simple No.
Flutter is designed to be multi-platform technology with a focus on mobile apps. Google Pay for example is one of the most complex apps ever built in Flutter but it is only meant for iOS and Android platforms.
In my opinion, the Web is more of an afterthought in Flutter.
Having said that, Flutter does have certain strong points. One is that if you are making a multi-platform app and you do not want to go through the trouble of having a separate codebase, Flutter isn't a bad technology per se.
I do not know any major company using Flutter for the web, however, the examples I am aware of are mostly those which could have very easily been web apps. Design IDEs and such. Flutter isn't a bad technology for them.
You can find a more detailed analysis of React vs Flutter web here.