There are several problems with the current chat:

1. The chat window (when expanded) takes up an inordinate amount of screen real estate when playing in a browser. This covers a large part of the map and makes clicking on territories a big irritation.
2. The use of tabs makes it easy to accidentally post on public chat when you actually want to post on team chat.
3. You can't tag a player in chat (e.g. @Boer) like in Discord/WhatsApp or other chat apps.

Suggested fixes:

1. Move chat to a left sidebar rather than the bottom. Chat posts tend to be short, they don't need a lot of width. There is always a lot of space available on the maps I play.
2. Make them never cover the map.
3. Remove the tabs and make public chat always on the top, with team and private chats below in that order. That will make it much harder to accidentally post in the wrong chat.
4. Enable tagging of players in post by @mentions

I think these are reasonably simple fixes that would add great value to players.