Flybe have been taking MPL cadets for years..
Whether BA runs their own MPL courses or not is irrelevant for recruitment. They accept MPL licenses and so you can make a move, if you desire, from EJ to BA whilst you don't yet have an ATPL. You are, by no means, stuck for your whole career as someone alluded to.
The MPL vs Traditional route has been much debated. There are pluses and minuses of both.
Ultimately you come to an airline from both routes with minimal experience. The key to success as a brand new cadet in an airline is down to attitude and your willingness to listen and learn NOT whether you have an MPL or a CPL.
(And yes, we see a number of folks coming into the airlines with 140 hours who think they already know everything. It's been both MPL cadets and fATPL cadets. And these are the ones who have issues).