Watching those clips the engine isn't under any load at all, just like reving in neutral.
It does sound like 6th isn't being selected at all.
Yes. Watched them a few times and, together with scottio’s description of what happens, I’d agree that 6th is not being selected. Points to a linkage problem, either inside or outside of the gearbox. You need someone to look at it who will spend the time to do a proper diagnosis, before taking stuff apart. As the saying goes, it’s not rocket science! So far you have been very badly advised and, frankly, taken to the cleaners for unnecessary work. From here, you need to get the thing fixed, before selling. The going rate for a car with an undiagnosed missing 6th gear will be horrendous!