I assume that the problem is synchronizing your Steam game account data with the game files. I have never deleted the application, so I cannot say this, but this is exactly the thought that comes to mind according to your description.
It would be best to be careful when deleting data, especially from the hard drive. My friend somehow accidentally formatted the disk when he just wanted to clean it of garbage. I had to contact
HARD DRIVE RECOVERY. It was possible to restore the data, but no one will return the nerves spent.
And why did you decide to delete Steam? It's a convenient platform for buying games.