I'm a little confused. Isn't this what babies do naturally? When DS was born he would wake (screaming like he was being murdered or dying of starvation), eat, play for a while and then sleep. Repeat q 3 hours. It never occurred to us to change anything - we interacted with him when he was awake. When he was tired - he fell asleep - anywhere - stroller, car seat, floor, crib, being held.
My son did not really 'play' when he was very little - just be awake a little bit, look around, move his arms and then conk out. Since from birth to six months, they should sleep 15-16 hours a day, the awake time your baby has is probably fine. If he wants to sleep, I would let him sleep. DH and I had a mantra - 'never wake the baby' - it worked for us.