I think, to be frank, that these boys and your husband have got used to being able to have everything their way whilst you twist yourself into a pretzel to enable it.
Your ex needs to either have 50/50, which means doing ALL the work including ferrying them around on his time, OR pay maintenance- he cannot have it both ways where he claims he has 50/50 but you show up to do the hard bit for him.
Your children need to grow up and start fending for themselves for a few hours at a time, especially if they have refused to co-operate with an au pair. An au pair for children in their teens is a bit nuts anyway, considering they may only be a year or two older than the children they are "caring" for!
They also need to accept that their extra-curriculars are no longer viable, not least because of the wholly unreasonable stipulation that parents be present at all events. As others have said, this must be a very niche sport considering I've never heard of such a bizarre rule- my child's clubs are all drop offs and she is 4!
And you need to stop trying desperately to make something work which doesn't work now, won't work in the future and by the sounds of it, never worked in the past, and accept there will be some fall out because all four of them have been spoiled rotten.