Luton was consistently the worst airport in TUI customer service questionnaires and became increasingly expensive and inconvenient to operate from, whilst demand for holidays from there continued to fall. Selling holidays with easyJet doing the flying allows flexibility of more regular departures and holiday duration whilst putting the operational burden of Luton onto a different airline. The fact the HQ is in Luton isn’t relevant, the two aren’t linked. One might argue the base only still exists because of the maintenance facility and the ability to rotate aircraft through Luton. It also exists to provide guaranteed lift for the cruise ships, so flights to Palma, Corfu and Tenerife (winter) will almost certainly remain.
Whilst easyJet also have a large presence at MAN, LGW and BRS, the demand for holidays from these bases is far bigger and there are significant benefits to having your own airline where the demand exists. Having said that, TUI do use easyJet from these bases too, often to plug gaps in days where TUI don’t operate to certain places, but also for the entire program of some destinations - generally places that are too expensive for TUI to offer a large holiday program, but require the flexibility of multiple weekly departures and different durations - Mykonos, Santorini (from next year), Malta and Madeira (from next year), for example.