Yes, it would be cruel to muzzle the dog and turn it loose with other dogs. How would you feel about being placed in a group of strangers with your hands tied behind your back, and not knowing if one of them, or all, were going to start hitting you?

Not all dogs can handle a dog park, and I wouldn’t stress out your male dog by taking him there when other dogs are present. Is there a down time when the area is deserted and you can let your 2 dogs play with each other? Otherwise, I would not take the male to the dog park and find another way to exercise him (long walks/runs, teach him to run alongside of you on a bike, or teach him to pull you on a bike or a dog scooter).

Some dogs are ‘mouthy’ and lucky for you, your dog knows how fragile humans are, hence the gentle pressure. It’s called bite inhibition (rather than bite prohibition), and it’s something all dogs should learn. If it bothers you, or he accidentally applies too much pressure, usually an ‘Ouch!’ or a yelp while removing yourself (stand up and/or turn your back on him) will stop him. Offer a toy for him to grab instead of your wrist. I had a ‘mouthy’ dog (lost him to cancer 1-1/2 years ago), and he learned bite inhibition as a puppy. We’d wrestle and play and he’d slobber all over me, but if he got too rambunctious and grabbed a sleeve/arm/gloved hand too hard, a simple ‘ow’ made him let go immediately. Then he’d apologize with a few licks, and we’d change the game to something else like fetch or tug.

I agree with safety reasons, plus, heaven forbid, some catastrophe occurs (fire, dog escapes, someone claims it’s THEIR dog), there is a photo of you with your dog for proof of ownership. Most people who deal with finding lost or stolen pets say you should always keep an updated photo of you with your pets on hand.