Feeling Stuck

I started potty training my 3 year old a week ago. I feel like it's going okay, but we haven't really had much progress.

I set a timer for 30 minutes to an hour. As soon as he hears the timer go off, he drops everything and runs to the bathroom and sits on the potty. I usually have to tell him to push some pee out. He wipes and flushes himself.

BUT, if I don't set timers, and just ask him if he needs to go, he always tell me "No." If I don't take him at least every 30 minutes, he will just pee on the floor/in his pants. He doesn't seem to really care at all. I've had him naked, commando, and with underwear, and all will stay dry all day as long as I have the timer going.

I make it so that he has to clean up any messes he makes, and I prolong it so that it takes longer than actually going to the potty himself. He knows it's wrong, as he is constantly repeating me - "Pee belongs in potty, not on floor or pants." And he will say "oops" if he pees on the floor. (pants he doesn't really care or notice until I ask him if he is still dry).

So, I feel like we have been at this place for a bit now. Does it just take time to initialize himself? Will he ever care enough to tell me? I'm not really sure what to do to help. I've read online that I should stop with the timer and just ask him every so often, but thats always a "no" and an accident a few minutes later.