I absolutely hate it when trainers or behaviourists take someone's dog off them and go about giving a display of how good they are for them. It's deliberately designed to make you feel inferior so that you'll pay for their services.
Yes, the dog probably does behave better for them, partly because they are an unknown quantity that is behaving in a very confident manner and partly because they probably do have years of experience reading dog behaviour.
As Idir said, her dogs will have met countless other dogs with varying problems and would have known how to give the appropriate calming signals to allow Jas to relax and start interacting with them.
Also, he may not have as much confidence to react if you aren't the one on the other end of the lead. I'll never forget when I took my FA boy to college with me and the tutor took his lead from me and asked me to step away. He literally changed demeanour in seconds - because I was no longer there for back up - he let her touch him and walk him around the class, he even sat and laid down for her ... and this was a dog that was usually seriously fear aggressive to humans.
Unlike your behaviourist though, my tutor explained why my dog was able to be handled by someone else in my absence and it helped us to realise that he was better off a the vets if we left him, as after we left he was meek as a mouse with the staff, whereas while we were there he was defensive and aggressive. It also meant we were able to get the occasional break by leaving him at a boarding kennels that was willing to work with us and keep him in a separate kennel block to their other charges.
I think it's not a case of you needing to be firmer - more you need to find a way to reduce your own anxiety around Jas's behaviour. I know, that's not an easy thing to do, but if necessary - take a training break and just enjoy Jas around your home and garden for a while. Lots of training and fun to keep him busy, without the stress of worrying about how he's going to react if another dog suddenly appears. I know you'll probably worry about him not getting his walks and freedom etc, but you have to remember that, just like our dcs, our dogs need us to be in the best state of mind we can be in, in order for them to take their lead from us. It's not going to be forever, but it might give you some space and ultimately perspective on it all.
Alternatively, if you do feel you have to keep pushing on - please at least have a chat with the two people I suggested - no obligation to go with them, but just see the difference in their attitude and the potential for how they could help.
Unlike others, who have probably never had a fear aggressive dog, I do understand your stress and anxiety and also the love of your boy that makes you determined to move heaven and earth to help him get past it. You're a brave and determined woman for keeping-on working with him and he's lucky to have you. ((have a very unmnetty hug))