Yes, for special teams' plays...kickoffs...punts...
Colts supply their balls for offense; Pats supply their balls for offense.
...the refs handled them on every play?
No. If each team supplied a dozen balls...and say, if there's another dozen special team balls...that's 36 footballs.
There's no way refs are handling all 36 balls on "every play."
What's your point?
So the teams place the balls at the line of scrimmage themselves?
If I’m a kicker, after 45 points, I might “let” a ref hand me the ball.