What I know of the intricacies of awarding Mail contracts to liner companies would fit on the back of a postage stamp. (Remember postage stamps?). And, to be honest, I have little intention of investing time in correcting my ignorance. Life's too short.
But my take on it is that the reason QE2 was never awarded a contract to carry the mail was that she was ineligible: only on a fixed run for part of the year. Why on earth would the GPO consider awarding her a mail contract? OK, she was on a bus run Southampton~New York, but then for 6 months she tootled off to the Caribbean or other warmer climes. Basically, I suggest the first test is you need to be a liner to gain the Royal Mail contract. Hence, of course, RMS Queen Elizabeth and RMS Queen Mary.
And you don't have to be a passenger liner: when I worked with Union Castle all their passenger liners, and a few of their cargo liners, were RMS. (But not the SS Reina del Mar, which spent its year cruising).
I would also suggest you would have to be UK flagged. (Neither of the two Safmarine-owned passenger liners managed and crewed by Union Castle were RMS - yet they undertook exactly the same run as their sister ships).
Most likely your liner run would also have to touch on UK on a frequent basis. (The public don't want to wait forever for their mail to be delivered, even if it is sea mail. After the mail bags were loaded at Cape Town the run to Southampton was about 13 days at the most, from memory).
So whether the Cunard management of the time wanted to distance themselves from past Cunard liners, consciously decided to start a fresh leaf, or whatever, I think is immaterial. SS QE2 would never have been awarded a Royal Mail contract by the GPO even if Cunard had pleaded for it.
Happy to be educated if my views are incorrect.