2001 sliverado 1500 5.3 starter clicks rapidly and does not engage the ring gear. Installed new starter got same issue

Test the alternator,usually free at Auto Zone ect.

Did you have it checked to see how much voltage you had going to the starter, and if the teeth were worn. I have even seen fly wheels with bolts loose. And are you sure that the sound was coming from the starter. If you timing gear, or chain has broken your engine will spin real fast, because you have no compression heard the sound many times at my shop and even with my own car. To see if it is the starter take it out and ground it some where on the block, and put juice to it and see if the solenoid engages and gears spin. In the old days sometimes you had to get shims so that the starter aligns with the flywheel. And one thing make sure you get the right starter. Or take it to a dealership, and leave your first born and hope they know what they are doing. I have seen a lot of dishonest mechanics, but most know what they are doing, but some tag on things they didn't do.

Found the bendix was worn, replaced starter. The found the battery was deficient and would not hold a charge. Low amperage was causing the bendix on both starters to chatter. New starter, new battery, up and running.

Doesn't surprise me that he needed all replaced. But with out hearing the sound, I gave other options. I was a mechanic before electronic ignition and scopes to hook up to. I was taught to listen,smell and look and if you could diagnose the car before you could hook a scope to. That's all changed now, but I can still smell the newer cars if they are running rich. Some of the smells and noises can tell you what is wrong.

Rapid clicking sound when the key is turned is a weak battery, glad you found it out. Unfortunately the hard way.