Southampton and Bournemouth catchments when combined are big enough to warrant a major regional airport. However, the reason they currently do not have a 5 mppa airport on the doorstep is threefold: 1) Bournemouth airport is positioned too far from the main urban areas so will never be able to support high frequency regional connections, 2) Southampton airport is space restricted both runway and apron and can't (currently) offer adequate LCC operations, 3) both locations are close to Heathrow and Gatwick, but Southampton is very close - in fact it's closer to LHR than other other regional airport in the UK at just 60 miles.
The future for SOU especially post pandemic (not over yet!) is going to have to be LCC with improved runway length and additional apron space in order to address all 3 points above.
Till then it's status quo.