And now we need to define " Won't turn over " , I take that as the engine isn't being rotated by the starter motor when the key is turned to the crank position.

Given the alarm engages I'd find a way to disable it. My 97 SL320 had problems with the alarm randomly sounding when I used the key to unlock then opened the door. The ATA ( Anti Theft Alarm ) on my car is a separate box that when unplugged, disables the system. Or maybe it was the remote locking module not sending a release signal to the ATA. An any event, unplugging one of these took care of the problem.

You do you have a metal bit key with a single button?

Remotes generally need to be synced with the car for them to work. Make sure the batteries are good first. For my cars, I press the single button on the remote 2X then insert and turn the key to the run position.

For a multi button smart key I'd expect a similar procedure but check around first.