Preschool really starts around age 3... any "school" that proclaims othewise is simply incorrect. If you were to observe in these settings, you will find that they do a lot of "free play" with short (10 minutes) periods of guided activities.
Those guided activities:
- puzzles
- play doh
- stories
- "collages"- gluing scraps onto paper
- sponge/finger painting
- sorting/categorizing items (anything you find in your house)
The whole point of early childhood education is to get your child comfortable with the idea of structured activities and routines including cooperative play. You can make ANYTHING into a structured activity or a routine. At that age my son loved to bake with me. It involved early measuring/math, direction following and basic kitchen safety- and he got to eat the outcome!
Just a thought.... keep doing what you are doing and set aside a couple of 10 minute breaks during the day for tabletop activities. Most centers have 2 year olds in a toddler room, so please keep that in mind. True preschools will not take a child until he/she turns 3.
Next year consider having him attend two morning sessions. He'll be developmentally ready and he would have a blast!
BTW- secondary content teaching skills are entirely different than early childhood (especially with your own child)! Good luck!