SQL Server - Very high disk activity after running query?
I wanted to post the solution that apparently worked for my case. I took down the app for maintenance last weekend and did a rebuild of all the indexes of the database (through the SMSS UI) followed by a statistics update and I'm happy to say we didn't experience the issue anymore since then.

I had rebuilt statistics before this and it didn't help and if the problems were related to index fragmentation then the queries should be slow all the time, but for some reason, it worked in my case.

Thank you, everybody, for your time and suggestions!