I'd have thought the private garages would also be incentivised to find problems - that they can then offer to fix for an additional fee.
As it is, I think most garages that offer MOTs in the UK are fair and honest, as the test is relatively strictly regulated, but I'm sure people do get ripped off.
The Netherlands has private garages do the yearly recertification inspection (APK here). There was a recent rise in 'remote inspections' where the garages had figured out tricks to avoid spot checks. This involved tricks like 'file the inspection very close before 17:00' or 'file the inspection result right before an actual true inspection' because apparently spot checks look at the last inspection.
They noticed the tricks as patterns, and are handling it. My point is, there is an incentive for private garages to do fraud here.