I do think that adding the wood increases the heat greatly and does create a much, much better end product for pizza.
We usually cook ours in the oven at 500, but they never really got that 'authentic' brick oven texture or end result. Both hubby and I LOVE how this turns out. It does not taste smoked, but does have a nice fire roasted flavor.
We have also just done them on the grill before and really liked it, but the top didn't really brown, it just wasn't quite like the brick oven. Although, still better than the regular oven in many ways.
Okay, so all that being said... If I were going to experiment with cooking on a charcoal grill I would use chunks of wood on top of the charcoal, not chips. I think the chips would burn up too fast, create lots of smoke, burn out and not produce the heat you are looking for. I would definitely do indirect heat, preferably by using fire brick or a pizza stone that can take the heat.
The biggest challenges to overcome. 1) Opening the lid and letting all that precious heat out. Probably not too big of a problem if you are doing one pizza. 2) Checking on your pizza for doneness. Again, you would have to lift the lid and let out all that great heat that builds up in the dome. 3) Doing multiple pizzas would be an even greater challenge.
But it is absolutely worth a try!
I bet you could still achieve amazing results.
For me, I have been grilling pizzas for 20 years and have wanted a brick pizza oven for the last 5 years. This was a great, economical compromise. Plus I love the fact that we can move our 'pizza grill' to where ever we want it. It is also easy enough that I can do it myself, without hubby.
If you try it, let us know!