Tbh, if you, your family or friends cannot remember to keep things out fo the way of the dog (is it a dog or a puppy, if it's a puppy, it seems to have chewed an awful lot of things!) then you might find a crate a good idea and it will keep your dog safe. If your dog has chewed so many things, especially batteries, rather than being a laughing matter, you're quite lucky your dog isn't dead tbh.
You're right that crates shouldn't be used to 'control' a dog instead of training, they should be used as an aid to training. So, at times where you can't supervise the dog 100% (cooking dinner, in bed at night etc) the dog goes in the crate. By him only having access to things he can chew when you are present (and this should be most of the time if you have taken on a puppy) he will quickly learn what he can and can't chew. I only really use my crates at night or when I go out without the puppy, which is never very long as I devote my whole time to the dogs, but nevertheless, we have never had anything chewed up that I can remember, except for our older dog for some reason best known to herself, having never chewed a single thing, decided to chew our telephone up one day. 
But yes, you should pay for the damage to your friend's bag. Your dog, your respnsibility and seeing as you asked if you should pay for your dog's mistake, not your friend's, then presumably you don't think it was your friend's mistake at all and it cannot be your dog's mistake, since... well... it's a dog.
If your dog had damaged something in public the law would hold you responsible, I don't see why it is any different in your home.