With a lot of use there are many ways for a DMM to go south.
The leads
The lead-sockets
Internal connections -- particularly wire to solder...
And fried chips, displays...
Further, they DO have internal fuses. Corrosive opportunities also exist.
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Your tale of abused equipment is the primary reason why I started bringing my OWN 'company provided' tools.
Every time Company Tools showed up they were broken. What happens is the troops lie -- constantly -- about their tool experience. So their forman gives them a brand new tool and they promptly ruin it and then hide it from the foreman.
This happened on one of my jobs -- a Greenlee hydraulic punch ruined before a SINGLE Knock Out was performed. It cost the fool his job.
If he'd asked for help I could have solved his mistake in 10 seconds. Instead he tried to dis-assemble the tool while it was stuck in the panel can!
