Looks nice, I think those are indeed some or most of the important things.
Another important thing that's currently missing or not visible is the notification of new BB-mail or Forum reactions, I think it's very important to engage users with the rest of the community to keep it alive. Maybe even add a notification if a forum post is liked or reacted to a lot that might be interesting to a user (i.e. a post on the competition page or on a national forum thread). This could be a red number either on the social button on the bottom menu or on the bell in the top right corner.
I think it would be cool if more aspects of the game are put into notifications that only require attention once, like news/headlines, training update, injury update, economy update and maybe custom notifications for supporters (for example a player matching a search criteria is added to the TL, or staff member has reached XX salary, time to replace). You don't need to read these each time you open the app, just the first time and then they can be on another place to find if you're looking for it (just not the home screen).