You stated two things I agree with whole heartedly...
"Time and punctuality are not important to me."
"I can usually wake up 1-3 minutes before my alarm goes off"
I can even wake up within a few minutes of need without an alarm by setting it in my head.
Another thing I can do is tell time within 5 minutes without a timepiece...
A big part of accomplishing this is to start tuning into your natural surroundings. If you have pets, they instinctively know when you get up, come home, eat etc if you are in a routine; we can do the same... no clocks needed.