As for why more British managers don't go abroad for work, from the Football League down you run into the island nation mentality... it means learning a new language, getting used to a new culture, and getting used to a different footballing culture.
The three English managers that have coached my Portuguese team (CF Belenenses), only one (John Mortimere) learned Portuguese. With Jimmy Melia and Jimmy Hagan I don't recall have heard them speaking a word in Portuguese. Jimmy Hagan the only two words he would say were "No comments". All of them stayed several years (with success) in Portugal. Not only they have adapted to our culture but also brought new ideas, new concepts to the Portuguese football.
Maybe they are a little more "lazy" today, but English managers brought a lot to football.
Those three managers are all old school. In their playing careers, they played at the top level.
There is no question that England produced some very good managers, we still do but as players they are effectively barred from the top level and as managers they now ply their trade in the EFL and VNL. A few still get exported although the tendency these days is to go further afield than Europe: Napal, Guam, India etc and they are, in general, very successful ... but there is no movement en masse.
At the top level though, the evidence so far is that the pampered princes of the Premier League either leave the game, court the "meedja", or enter into a kind of voluntary zoochosis; exotic breeds thriving only in the rarefied stratosphere of high fantasy and camouflaged by corporate identity.
Football in England, for players, managers, and fans, is very much alive and well. And, as a bonus, we get to watch the pretties for a few minutes before the film starts.
I love the smell of TOAW in the morning...