I would hope that no groomer would hit a dog.
IME it's common for dogs to be pulled, knots to be yanked out, low level hit (not a full on punch). It's not right and I don't agree with it...
...BUT dog owners also need to look to themselves. Your dog groomer is there to groom your dog, not to train it. Most dog owners will slag off a salon behind their back, if not making an outright complaint to their face, if the standard of the end result groomed dog isn't top-notch. Or they'll tell their friends who their groomer is without mentioning how difficult their dog is, so people think the groomer produces shit work.
It takes two hands to do the work, one holding scissors/clippers/ angling dryer and one holding brush/comb. It's impossible to do this properly whilst also restraining a dog. If you're only willing to pay a cheap price for the job and you want it done well, that means it has to be done quickly otherwise the groomer can't make a profit, and this leads to dogs being knocked around.
If you don't want your dog muzzled, teach it not to snap at people. After all, you're not going to pay the groomer's wages if you can't work through injury are you? You're also not planning to compensate them for any life changing injuries or disfigurement they may receive due to your dog.
Regarding the cord restraints, there are dogs who won't even stand on the table and will insist on trying to sit/lay down constantly even when restrained, or be always wriggling around. They're effectively hanging themselves on the restraints, but if the groomer loosens the restraint they'll have a dog sitting/laying down who they can't groom. What do you want groomers to do? It's not their job to train your dog, you need to do this at home. Or you at least need to pay a reasonable price so the dog groomer isn't in a hurry and to accept a half-done poorer standard of work (and not to go slagging off your groomer because of it!) while they're in the process of gradually training your dog to accept grooming.
I don't know what they're charging these days but back when I was doing it, the going rate was about £20 for difficult (working on fidgets, the majority of customers), physical work (using safe and correct physical techniques to partially restrain a dog you're also trying to work on is incredibly tiring). At the time minimum wage was about £6.50/hr plus all the business related expenses. £20 doesn't go far.
I didn't beat up dogs at all which meant I often over ran on time and could fit less work into the day. It also meant untimely my business wasn't viable, so I closed. Customers mostly wouldn't pay a fair price for the actual work I was doing, the time it took, the skill I had in both training the dog and grooming it.
Anyone can clip a dog or cut out matts with scissors, it's not hard. Matts which shouldn't exist! Brush your dogs, people, daily/weekly according to coat type and use the correct brush, then check for bits you've missed with a comb and brush those places again. Don't present them to the groomer having not be touched for 2mths or longer and expect a brand spanking new perfect haircut for a cheap price!
Trimming a dog neatly and tidily and clipping to particular styles is a skill and is largely what you're paying for after the general business overheads of water electricity shampoo specialist equipment and maintenance, rent and rates etc. Goomers aren't there to cheaply deal with your untrained, unbrushed, snappy dog because you don't want to.
OP the groomer should (not legally required though) have both public liability insurance and professional negligence insurance (care custody and control). Its not like car insurance. If they weren't negligent the insurance won't pay out. It doesn't look good that they're trying to cover up what happened, but that still doesn't mean they've done anything wrong. It could have been an accident eg dog falling off table (even if restrained, you'd have to cut those quickly so the dog didn't literally hang itself and die, so there'd still be a fall just a reduced one). Doesn't make it right, but in today's society where people's DC and pets can do no wrong in their eyes, many quick to dish out verbal abuse and become aggressive, I can see entirely why a groomer may fail to mention an accident took place. I wish you luck with getting the situation resolved and finding answers to what happened OP. I hope your dog makes a full recovery.