Enforced naps during day

We have our pup exactly 7 days today and he also just started 13 weeks. We have his pen and crate inside set up in the living room and he eats every meal in his crate, he goes there sometimes during the day or he goes to his crate when he gets scared by something to chill there, but he doesn’t nap there. I started closing while he eats or live the room and got to about a minute so far before he starts protesting. I wait for silence and then go back. At night, he sleeps in his bed on the floor in our bedroom as he asks to go outside for a potty and wake us up when he needs to go.

I work from home and two times a week from the office, my partner works fully from home. When I am home, I enforce his naps in my home office in his bed, he tried to fight it sometimes but then eventually gives in. After work, me and my partner usually chill in the living room and just wanna cuddle with him too, so he’s with us on the sofa. He naps a lot there, he loves watching TV and he falls asleep while watching. Best to make him sleep is Bluey lol he usually doesn’t follow us when we get up and go somewhere, he will sometimes lift his head to check and go back to nap. When he does follow, he checks what we are up to and goes back.

When I am in the office, my boyfriend works from the living room instead of his office so pup always nap on the sofa with him. He kind of just fall asleep on his own. He does follow my boyfriend everywhere on those days when I am not home and checks my office few times to see if I am there. Since we got him, I was in the office only twice.

He plays with his toys by himself, he goes to another rooms and eats without us in the room too. We still watch him so he doesn’t bite a lot of things.

Did I make a mistake not pen or crate train him properly to nap there? Is it too late now as he has this routine? Or at 7 days in the house is still ok to start training? I want to pen train him to be able to leave him in the house alone when we go shopping etc.