He does sound to have settled in well - the Barney house is a good size for a nest and a hoard so can see why he chose that.
I wouldn't wake him just now and give him 2 or 3 days to do his own thing, but if you see him offer him a treat through the bars. I find they take the food from the bowl, hoard it and eat from the hoard while in their nest
In privacy. Fresh food is maybe irresistable so eaten there and then usually!
If he shows signs of wanting to come out in the first 2 or 3 days then no harm in that but they can get freaked out easily and have set backs so 2 or 3 days is a good idea to leave them alone. It takes about 2 full weeks for them to be fully settled and confident in the cage so ideally no cleaning during that time except emptying a potty after a week maybe or spot cleaning a handful of pee'd on substrate after a week (or longer) if he pees in a place other than the potty. And then move the potty!
And then minimal cleaning for as long as possible! Sorry I'm going on again. As Souffle says making a bit of cage noise at feeding time lets them know it's feeding time and can wake them and feeding around the same time each evening is a good idea. Then they get in the habit of waking at a certain time. When they want to come out can vary with the hamster. Both ours have always gone back to bed with their stash after feeding, but come out again later.
With our last Syrian Charlie he would never have come out at all if i didn't wake him and coax him out. He was a home bod! I found 8pm to 9pm he was quite dopey and easier to handle (or would just curl up and go to sleep!) but very active late at night and very active around 5pm or 6pm. With a 6 year old you want them out earlier than 8pm really so feeding around 6pm might be good.
With our current Syrian Newt he shows he wants to come out regularly - and that tends to be when the food is put out, but he comes out first then goes back and pouches everything after. If he doesn't want to come out that night he just takes the food and goes back in his house. But then sometimes wants to come out later.
So it's partly playing it by ear with the hamster's personality and partly a bit of training