A teacher trainer can often just be temporary and the increments aren't worth the ulcers.
Yes. Every teacher trainer also teaches students. There is a very limited aount of work doing that and a lot of people chasing it.
I share the same reservations about the British Council. I have no room for elitist or dry folk and they may fit that description.
Maybe some do, but by no means all. Though you can certainly find that.
The Council is scaling back operations in Europe though, so it isn't the stable job some people think - some time ago they all had to reapply for their own jobs, and the staff levels in PL are but a shadow of what they used to be, the salary seems good, but locally recruited staff get a lot less than centrally recruited staff, and even then a good teacher can more than match that working freelance.
And they all seem tired and stressed due to the long hours, particularly the non-teaching time during which they have to be present and available.