Why?
Hmm.
I hope your joking there.
Yes, I'm joking. or am I (strokes cat)....

Why?
If they are a good franchise who thinks it will make money they will run it well, if not they won't. This sounds like some kind of WR hates SR and visa versa, whice I cannot believe is still so true that a sensible management who wanted to make money and valued the route could not overcome.
If both lines are under the same management, the management will concentrate their efforts on the current main intercity route to Paddington, they will welcome any revenue the Southern route brought but you would find an end to attractive fares to London on the Southern route, Waterloo trains caped at Yeovil Junction when GW trains are diverted that way (which used to happen under BR and in the early days of SWT until SWT put a stop to it - meaning longer journey times for diverted GW trains and fewer of them) etc.
The management just won't have the time for much initiative on a "secondary route" and will find it very tempting to borrow some of the 159s to overcome the famous GW stock shortages elsewhere, leading to unreliablity and overcrowding. You certainly wouldn't see initiatives such as the proposed trains to Yeovil Pen Mill.
Do you think any of the Evergreen upgrades would have happened if Marylebone to Birmingham had been part of the West Coast Main Line Franchise? The route would still be largely single track with one an hour venturing north of High Wycombe and terminating at Banbury.
It would also be disastrous for operational reasons, meaning more than one operator into congested Waterloo (which is why the plan to move it to Wessex Trains got canned) and disrupting the current successful SWT NR alliance and amounting to an Orcats raid on SWT between Basingstoke and Waterloo.
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