Why would it be a problem for one TOC to run both routes, but not for say, Cross Country to have the sole Birmingham to Leicester train, or London Midland the sole Birmingham to Lichfield trains? I don't know how concepts of unfair competition can exist on something like rail where there are typically only a few operators anyway.
I think the "competition" aspect gets exaggerated - there are several "point to point" journeys (run by a single TOC) which are bigger than the entire "London to Scotland" market.
For example, there is no competition on London to Manchester services.
There have also been a few examples of one company owning two TOCs in the same area (Valley Lines and Wales & West, Central Trains and Midland Mainline, NXEA and C2C, Connex South Central and Connex South Eastern...), which suggests that the "no one TOC can have a monopoly" argument is more a hypothetical one.
There will always be one TOC with a monopoly on some routes, its a fact of life, but there will always be more competition from cars/ buses/ coaches/ planes...
I think it'll depend what the next franchise is called. The Cross Country franchise is called CrossCountry not Arriva CrossCountry. Virgin used the name Cross Country in some instances anyway e.g. on their route maps they referred to Cross Country routes and on rail travel vouchers they issued for delays on Cross Country routes they said Cross Country Trains Limited. Therefore, a lot of people could think the change in name is just a rebranding exercise by Virgin.
If the new West Coast franchise is called West Coast Trains the same thing will happen. If it's called First West Coast, Arriva West Coast etc. then more people will notice the change in company operating the service.
There was someone on here earlier this week asking about when "GNER" would be getting new trains for the East Coast franchise, apparently not appreciating that its been a long time since GNER operated that service.
People in Sheffield still refer to London services as the Midland Mainline one - it takes a long time to change perception, and I don't think the "average" passenger pays too much attention to that kind of thing (as long as their train turns up).