I'm still on 3.12 and Data Source Manager is working for me - but its probably because im not connecting to the same data sources as you. The 'Getting Tiles' message at the bottom suggests that you have a connection to 'something' - a Tile service? in your project...
Your problem is more than likely related to a service that you previously had connected in your project which - for some reason, 3.14 doesn't like.
You could use the 'Browser' to view all the services and data sources you have in your project there. This may help diagnose your problem. You should be able to remove particular data services from the 'Browser' toolbar, and then try opening the Data Source Manager. Once you find the service that is causing the issue, maybe try reconnecting it/adding it as a new service/connection definition from Browser.
In order to debug and remove a service you think is problematic, do a really good 'cleanup' of that service definition...….. . That is - Delete the connection from your Browser, save the project, then re-open the project. Don't just delete and re-add. Ive found QGIS 'hangs onto' service connection definitions somewhere, so do a really good cleanup of the problematic service definition. (This may also include removal of layers that use that particular service).
Obviously maybe make a copy of your project for debugging purposes!
