All season tyre are typically directional. ie they do need to to go on the correct way round to get maximum traction. You should have had two tyres mounted for correct rotation on the left side of the car and two for the right side. If you look carefully at the side wall there is normally a 'direction of rotation' marking. Usually in words but it might just be a stylised arrow design.
If they have been mounted on the wrong side ,get them to swop them over. If they have mounted the tyres wrongly (eg all 4 the same way round) they will need to remove and remount the incorrect tyres. Either way its their mistake and they should do it free of charge.
As a great fan of all season tyres I would be very surprised if ,correctly mounted ,any brand would give worse grip in mud etc than summer tyres. I drove in snow on mine on 5 November 2023
But bear in mind if your car does not have traction control it is in effect just one wheel drive. Two wheels can potentially give drive if the tyre grip/road surface is more or less equal both sides. But if the grip is unequal , maybe one wheel in mud or ice,but the other on good tarmac drive will follow the path of least resistance. The wheel with best grip will remain stationary, and the car may 'spin' the other wheel and you go nowhere. If your car is on a slide slope the lower side of the car will be heavier. Even if the low side is the one in the worst mud or snow, it may be the lighter, high side, of the car that spins its wheel first.
Even with the best possible tyre grip in mud or snow, this can still happen. The car may be unable to pull itself out of a ditch etc if its effectively relying on traction from just one of its tyres, not both. Even if both wheels are potentially able to offer similar grip, climbing a steep enough slope or hill in snow can result in one or other of the wheels 'spinning' before the other. All season/winter tyres give you a bigger margin before this happens but all tyres have their limits.
You can improve traction by technique.. a wheel is more likely to 'spin' if you try to accelerate too fast. Grip may improve if you attempt to crawl out in a higher gear. I used to compete in off road motorsports
attempting to get a car as far as possible up muddy (sometimes icy) tracks for fun. Attempting to 'blast' your way out with wheel spin often doesnt work , however good your tyre grip. The drivers who perfected the 'trickle' technique usually did better.
