It would be reasonable to expect that any considerable increase in traffic at SOU for EZY would risk cannibalising their LGW operation so if they do anything, they'll need to tread carefully, especially given that BA have considerably increased capacity at LGW this summer with the slots they purchased from Monarch.
I thinks it's also worth considering that whilst SOU might benefit from more European destinations from EZY, domestics would likely lose considerable frequency. You only have to look at IOMLGW which, when competed, went from 3 / 4 frequencies with Flybe to just single daily on some days with EZY. Not much use for business. Now that may not happen at SOU but there is a reason EZY haven't decided to build an operation there already.
If services are to be launched and sustained, you need an airline that wants to do so and an airport willing to support that development. I wouldn't default to the view that the lack of development at SOU is solely the fault of the incumbent airlines, There must be some reason as to why Flybe have reduced the number of based aircraft at SOU between last summer and this summer whilst maintaining or growing their numbers at other bases. If the environment doesn't facilitate growth, that's not necessarily the sole fault of the airline and they shouldn't be blamed for it.