There should be a career path which could allow a cabin crew member to be on the Board of BA.
I have worked in airlines where Flight Ops and IFCE are separate entities like BA and also in ones where the two empires are unified under a Director of Flight Ops. The DFO then represents both FC and CC interests to the Board. The unified approach is by far the more operationally sensible. There are no occasions where the CC managers order a diversion or decide when the CC need extra nights off or indeed anything else because they work for the Ops department and have a responsibility to the safe and efficient conduct of the operation. Simply put it works better this way.
There is no reason why a CC member should not rise to be the DFO - all they would need to do is understand the operational side of the airline. In the more dysfunctional BA model the CC generally have a great understanding of the IFCE side of things but generally a very poor grasp of operational matters. This is largely due to there being no need for them to have an understanding of the more technical side of the airline business - it certainly does not imply any lack of ability, more a lack of opportunity. Similarly the FC community are generally less aware of the day to day stuff that CC have to deal with for the same reasons.