03 lexus ES300, Alternator belt broke off
 

2003 Lexus ES300. Before the incident mentioned below, the car was in good shape and no sign of any issues affecting the drive.
Hi everyone.
My son got a host of engine warning while driving in freeway but he continued driving it for miles until everything completely shut down. He called and I went and took a look. The Alternator belt had broken off and missing and the battery completely depleted. We had to tow the car home. Next day, I installed a new belt and a new battery ( the old battery was no longer chargeable). I ran it in idle for awhile and readjusted the belt tension the best I could and checked it manually by making sure the center of belt can only turn half way by the force of my hand. (repeated this process a couple times). Now, the car still starts ok, but when I put in Drive, it shuts down and there is no engine warning.

Is this still due to the belt tension not being correct?

Or

Other damages are done to engine because of the stress he put on the engine for driving it without power generator until everything shut down?

Thank you in advance for any constructive help.