Many on here feel that it is 'wrong' to need to fly an aeroplane registered in another country with a licence issued in that country. It has been that way in some way or other for over 100 years. If that is 'bothering' you, you just need to get used to it. That is the way things are.
However, the question that this Thread is all about is "Are UK airlines pushing for the UK CAA to recognise EASA licences."
My answer to that would be that airlines employ pilots who are licenced to be able to fly their aeroplanes. Airlines don't licence pilots, they expect the pilots to come to them licenced. (Unless of course they have programmes to train pilots. but in that case it will be tailored towards the licence that they require.) So, why should any airline be pushing the UK CAA to recognise any other licences? It is entirely up to the pilots to ensure that they have the licence required to work for any airline. Should "UK airlines push for the UK CAA to recognise" any other licence? That is the pilot's "problem", not the airline's. The regulator sets the requirements, not the airlines.