The Merchant Navy operators get a reduction in Tonnage Tax for employing British officers. With flags of convenience and crews from the cheaper parts of the world becoming the norm, the government reckoned that if the trend continued there wouldn't be any British officers left. Therefore they brought in this initiative to act as an incentive for firms to employ British officers.
Additionally, it's almost impossible to train to become a Merchant Navy officer whilst sailing aboard a motor cruiser or similar, you need time on-board larger vessels and the only way to achieve this is with a sponsoring company.
Personally, I'd like to see the government reduce APD for airlines above a certain size which fund cadet programmes up front/guarantee a loan and increase it for those who don't. Of course it would only apply to the big boys, not the smaller TP operators etc. who are no doubt squeezed anyway.
The Train Operating Companies often get 300 applicants per vacancy for train driving roles but training is paid and you get a very good salary during training. That said, the railway tends to be fairly left wing politically and the cultural norm is for trainees to be paid.
Aer Lingus most likely chose FTE because all of their cadets go there, the school is of a known quality to them. Wouldn't do untagged integrated myself though, huge risk.
As for bureaucracy, the CAA have of course been involved in the Red Tape Challenge in relation to GA, hopefully we'll see this reduce costs for training outfits, wouldn't bet on it though.