School can be part of the problem, as child after child comes down with something. Whatever it is spreads and then often makes a second round.
Tonsils are part of the lymphatic system that fights infection. That's why taking them out is as a last resort. Your son's tonsils are enlarged because he's fighting some kind of infection all the time, sounds like. It's good you're going to see a doctor - maybe more than one for a second opinion - to get him properly evaluated.
Maybe he hasn't fully recovered from something? Maybe it's something that has gone undiagnosed?
In the meantime, what time does he go to bed? 6 yo and sick a lot may need to be in bed MUCH earlier. Think 6 yo = 6 or 7 PM is bedtime. Then when they're 8 or 9, they can stay up until 8 or 8:30 pm, 9 pm on special nights. Then as teenagers, 9 or 10 pm is no problem. It keeps them healthier and well rested. YOU, as well!
Some kids, and adults, just need more sleep. Functioning tired makes you more prone to illness, child or adult. Are you confusing a child's "wired-but tired" energetic nature and desire to learn... everything... with true energy?
People - and kids are people - get used to functioning with too little sleep, and not feeling tip top, they learn to ignore the messages their body is sending them.
We may not go to bed truly early often in our "modern" society, but our bodies don't know the difference between living now and two hundred years ago. It's FUNCTIONING hasn't changed much - only our technology!
Remember, Benjamin Franklin: "Early to bed, early to rise, makes a man healthy, wealthy and wise." It was just easier to go to bed before electric lights and computers, TV...! Good luck - I hope he feels better.