when companies overhaul....

a diff or an IRD on the bench...do they run it up at speed and under load to check for whines on overun etc....or just stick it on the shelf for sale?

Straight on the shelf i'd imagine.
If they're confident they've replaced everything correctly why (and how) would they load test it?

When I was on the tools, some years ago, one of the jobs was overhauling commercial vehicle diffs and gearboxes and we never used a test rig (only used on engine overhauls)
If the IRD is rebuilt correctly using the correct materials and tolerances there should be no need to run it on a test rig.

Saw a program on discovery channel once called "A Car Is Born" or MG or 4X4 is born - there were a few similar ilk & they visited a transmission rebuild shop who rebuilt overdrives amongst other things for classic cars. They had a test rig which they bolted each unit to once rebuilt & ran them up to speed before even considering selling it. To be fair the overdrive is quite a complex unit but I suppose if the reconditioners are high quality old school engineers then they wouldn't have it any other way.

That was on an MG is born, I bet he paid extra for getting it on the test rig