He rarely naps, loses interest in a chew within 15 minutes and then finds trouble. He becomes a cranky, overwound puppy that cannot settle. Our trainer discourages use of the crate for "settle down time" or timeout, but we feel that the pup needs some way to settle down and he's not ready to do so through training.
I strongly disagree with your trainer. Some down time in the crate is exactly what a cranky, overwound puppy needs. Cava is only 5 months old, but I've done the same with every previous puppy. Some people say using the crate for timeouts will make the dog view the crate as a negative, but that has never been my experience. It's not necessarily a punishment, it's just a break. Both for the puppy, and for you
Usually when our puppies have gotten to the point where they're just bouncing off the walls being totally annoying they're minutes away from crashing. Cava gets these periods in the evening shortly after dinner - she'll be wild eyed and panting, sometimes doing zoomies, and nothing works to get her to settle down and focus. We put her in the crate and in no time she's out. She sleeps in a plastic crate in the bedroom but we set up a wire crate (you could also use an x-pen) in the living room, so she's with us and the other dogs rather than isolated from the pack. Sometimes she fusses a bit, but rarely for long. He may shriek if he's not used to it, but simply ignore him. Don't talk to him, don't look at him, he does not exist. You can put a bone, a chew toy, or a stuffed Kong or something in there with him, but once he goes in he's invisible until you're ready to let him out.
We are beginning to wonder our pup is not responding to the purely positive training methods. In the past, we were willing to handle counter surfing, chasing our cats, jumping/chewing on furniture/people, and general misbehavior with corrections. We used leash correction, shaker cans, squirt bottles, etc. and our dogs learned quickly. Now, we are encouraged to ignore misbehavior and wait until he makes the correct choice. It doesn't feel like the approach works for a big, active, young GSD.
As much as possible, I prefer to teach my dogs what TO do and then reward them for doing it. The more I can reinforce good behavior, the more I get good behavior and the less I need to correct them for the wrong behavior. I also like to allow my dogs to learn to make good choices. BUT, some behavior is self reinforcing, (chasing the cats is fun!) so unless you can manage the environment to prevent him from practicing bad behavior, then correcting him is going to be necessary. Otherwise, he'll just keep doing it. Attention barking can be ignored because if it's not working to get attention he'll eventually stop and try something else. And while annoying, barking not harmful to you, your cats, or your guests. Jumping on people isn't going to extinguish on its own, so management, training an incompatible behavior (he can't jump on people if he's sitting politely for greetings), and sometimes corrections are all appropriate.
If I'm in a class and they do things differently from the way I train, I just continue doing what I do because I know it works. Class is only an hour a week, so most of the training happens between classes anyway. If you're working with a private trainer and don't think their methods are working for you, I'd look for a new trainer.