In the context of a diesel repair shop, he likely was referring to fork lifts or vehicle lifts rather than elevators.

This doesn't require an internet, just a LAN.

It's a car lift. Not only would it be irresponsible to rely on a computer to tell you when you should maintain it, as some inspections can only be done visually, it seems totally pointless as most inspections need to be done manually.

Get a reminder on your calendar to do a thorough inspection once a day/week (whatever is appropriate) and train your employees what to look for every time it's used. At the end of the day, a car lift on locks is not going to fail unless there's a weakness in the metal structure, no computer is going to tell you about this unless there's a really expensive sensor network and I highly doubt any of the car lifts in question have such a sensor network.

Moreover, even if they did have such a sensor network, why are these machines able to call out to the internet?

These requirements can be met by making the lift's systems and data observable, which is a uni-directional flow of information from the lift to the outside world. Making the lift's operation modifiable from the outside world is not required to have it be observable.