DGR - yes bonus for Irish and also Scandinavians who tend to have a high standard of English. I know this from people doing TEFL teaching in various EU countries. Who is going to bother sponsoring people for a visa, and waiting until they get it, if someone else can have an interview on the Saturday and say that they can start next week? (A real life example there.)
I don't know, but I would suspect that if the UK had gone into the negotiations without the belligerent attitude that they still have, that some mutual agreement could have been sorted out, as with musicians.
Yes, there are also plenty of TEFL jobs in China, South Korea and South America - but they are all a long way away for a UK based student. Strangely enough, this is one advantage Covid brought, with teaching switched to on-line - Skype, zoom, etc. so now language teachers don't necessarily need to be in the country where the student is.
I feel the same sort of anger for people who had thriving export businesses to the EU, who are now told to trade with Canada or where-ever instead. Despite whatever Johnson, Frost and all the other crowd of Brexiters think, the EU is part of the globe, and the UK won't be global in anything if it pretends otherwise.