I need sleep too!

Well, this is better than a human baby. You can be up all night with feedings and changings for well over a year. Not that you're any less tired, but this isn't permanent.

At 6 months, he's just gaining bladder control and may not be able to handle 6-8 hours without going out. Also, this time of year, the sun is up and the birds are singing well before 6, that may well be waking him up and once he's up, just like everyone else, he has to go.

You can try pushing the wakeup back a few minutes at a time and see how that goes.

You're lucky. With our second puppy, we were still taking her for midnight bathroom trips until she was over a year old. She's a small dog, and couldn't hold it as well as our others.
As Amaryllis said, 6 months is when they first start getting complete physical control of their bladders, so he may be at his limit with how long she can hold it right now.

When he wakes you at 5 am, do you take him out? Does he really need to go? I mean, does he go as soon as you take him out, as if he really NEEDED to go? If so, I would say he's doing his best, and that's his limit currently.
If not, if he doesn't seem in a big hurry, then try pushing him back. Wait til 5:10 before letting him out. Stay at this time for a week. Then, try 5:20, for a week. Then, 5:30. You get the point, gradually, over several weeks, try getting him accustomed to staying in bed longer.

I would do the same, take him out at 5:10 but don't talk to him, no food or water, no praise or treats, one minute to pee if that's the issue then back in the crate till six. He'll protest a bit but eventually get the idea that it's not morning time yet. Putting some background noise in and blocking the sun will help too, if he's hearing traffic or seeing light. Not the instant solution but at least you can snooze a bit longer that way. Keep the five am pee break quick and boring!

Could you get up in the middle of the night with him and take him out then to see if he'd sleep later? Reason being if I was getting up an hour before my alarm went off I'd never actually get back to sleep!!

Some dogs seem to be ok all night, other dogs aren't. I've known adult dogs who couldn't hold it all night. He also may still be getting used to the routine. It took more than a month for our pup to really tune into the rhythm of the house. We lucked out and our pup has been good for 9 hrs overnight since he was 12 wks old. He hasn't had a single overnight accident in over two months. Last night I fell asleep on the couch with him and woke up at 4 am and figured I'd pop him outside to pee and he had no interest in going at all!! I had to put him on the grass with his potty command three times before he'd go, then he dashed right back into the house and went back to sleep in his crate.