Yep - a Sunday in August, on the Paris peripherique. Obviously.
However, the system clicked into working almost immediately. We'd had a front tyre blowout on the campervan (biggish motorhome), and the front underrider had also been damaged. My BIL had done an amazing job of keeping control of the vehicle, and getting us onto the hard shoulder. A wee man came along with a van, and cones, etc., whilst we were doing our bit with the warning triangle and the high-vis jackets. He spoke no English, but luckily, my French was good enough.
Our AA cover was also for the EU, so we managed to get someone to come out, and put us up on the back of a truck (as it was too heavy to jack up at the side of the road), and they took the 4 of us to a garage. It then transpired it wouldn't be able to be sorted until the next day, and as it was also our accommodation, we booked two rooms in a B&B nearby for the night - the garage owners wife sorted it for us, and was super helpful. I can't remember if we paid for the B&B (it was about 5 or 6 years ago), but the van was sorted the next day, and we were able to continue on our journey on the Tuesday. It was all managed well - and luckily we had maybe 3 or 4 days until our ferry home, so didn't need to re-organise our crossing.
Hope that it all works out for you OP. At least you've got the cover in place, and it's all sortable. No-one is injured, and you will all get home safely soon.