I don't think 20-30 minutes of playing/winding down by self in bed is a big deal. My 2 1/2 year old has always needed that long to get to sleep--even as an infant. He has never just gone to sleep in 5-10 minutes. He's always singing, kicking his crib, throwing things out, etc for about a half hour before going to sleep. It's not that he is not tired or that he doesn't need a nap--it just takes him that long to settle himself down. He sleeps for 2.5-3 hours, so I know he needs the nap!
My suggestion is, since you know it now takes awhile for him to fall asleep, to put him down 30 minutes before you want him to be asleep. I would shoot for 12:30 and if he is asleep by 1 or 1:15, that's great. If not, it's not going to hurt him to be awake in his crib for awhile. Some days, mine doesn't sleep for some reason. He still has to stay in his crib for the nap period. I usually do 1.5-2 hours if he doesn't sleep. I need a break and even if he doesn't sleep, he gets a break as well.
I don't exactly know what a sleep reminder is, but if it is you going in there an telling him to go to sleep, I'd skip it. I'd only go in if he is crying and needs some help settling down. If he is just not sleeping, I'd let him be.
I used to put my lo down at 1:00, but as he got closer to 2 1/2, he started taking longer and longer and longer to fall asleep--Sometimes he wouldn't be asleep until after 2:00. I bumped up nap to 12:30 and it helped a lot--usually asleep by 1 or 1:15 now.