So after 5 years I only used this idea once, it's unneeded for most work, I cut the original hole through the drywall, or 1/2" plywood sheathing (depending if I'm drilling from the inside wall out, or outside in). When the hole saw cuts through the drywall or plywood exterior sheathing, stop the drill, remove the plug unwanted donut hole material. Then reverse the drill, and the hole saw will not grab the fiber glass batt from the wall, and cut a clean hole through it.
In my case, I have a hole saw extension driver, as the studs are 2x6. Once you have kept the alignment to where your exit hole will be on the other side, the pilot bit should create a hole, and the driver tool can be reversed back to proper cutting of the hole saw to finish hole through the wall.
I'm mostly installing power and mini split refrigeration holes through my walls.
I will typically inject foam from a can via a long hose and a hole to inject it to reseal my intrusion of the vapor seal plastic sheet on my inside walls. And this has been used to fix when the fiber glass batt did get grabbed and spun around the hole saw.

The trick is reversing the cutter so it does not grab the fiberglass.