Am I overthinking

My boyfriend and I have overlapping schedules he works 7a-3:30p and I work 4a-12p. Which mean’s typically I have to go to sleep about 6:30-7p so time with him can be limited. Yesterday he got home at 5:30 from work and I asked him to spend the little time with me, he stated he wanted space, he wanted to play his game for the second week in a row. I have went to sleep at 6:30 every evening alone, I don’t mind if he plays the game but I feel for a healthy relationship we need at least some time together. I just wanted to go to sleep in his arms, that’s it. When I tried to tell him this he said “we need to be our own people. Find a hobby or something to do while I play the game.” While I agree that we don’t have to spend every second together, I just feel like he should want to see me more than if I get paid or he wants a new game. While he does express and show love to me sometimes, I feel like maybe he wasn’t ready for a relationship.

Edit: originally I was not clear, at 6:30 when I go to sleep I don’t expect him to go to bed with me. I want him to at least acknowledge “hey she is not going to be awake for long maybe I can watch an episode of a show with her until she falls asleep AND THEN play my game.”