IMO, and what I've found, is that a kamado / egg / primo etc... is much, much easier to produce a nice pizzeria style // NY style pizza than any standard kettle out there, because they do have a much more oven-like profile. But you can do a great pizza in a kettle with the right attachments...hard to keep the heat in those things though.
I have a Primo oval xl kamado, and using pure lump charcoal, I've made a number of pizzas over the last year. They're actually great--and no-one who has eaten one has ever complained! But they're not super high temperature pies. So I'm cooking them at 500-600F dome temperature over ceramic plate inserts. That means a 14" pizza in about 8-10 minutes (if you haven't overloaded toppings etc...).
I've been experimenting with a number of stones, including thinner and thicker ones, and the Big Green Egg stone. The best I've found by far--and the most versatile, because it's glazed--is the new Emile Henry BBQ pizza stone. It's a very interesting, very high tech, very high temperature, very thin device with extreme thermal shock capability and heats up very quickly too (they actually say "no preheat necessary"--and it does work, but not optimally). It transfers heat though extremely well compared with regular stones--and that's the secret to it. I've found you do need to turn the stone a number of times or it will find the hot spots in the kamado--even with ceramic plates below!
So here's what I've done so far:
1. build a full fire on the primo oval
2. once fire is going, add ceramic plates (this makes a radiant floor)
3. build dome heat to 500-600F
4. hold for 40 minutes to an hour to optimize burn rate. The good thing about a kamado is that to hold 500-600F you don't need much of a fire at all once the plates have heated up. You really are working with an oven at this point
5. pre-heat pizza stone
6. turn pizza once or twice (you don't have to do this more than once if you've done the pre-heat--if you're in a rush, well, you need to turn a lot, which can just make things worse).
This works really well--one large or two small pies at a time. You can even cook them with a bit lower heat and longer bake times. You'll still get some wood flavour, and of course you can use wood chunks or a log in there too. For the stone, for example, Emile Henry recommends 465F (but I like the results better at higher heat, especially with any mix of 00 flour).
Now, people do All Kinds of Crazy Things with their kamados to try to replicate a wfo and 800F + cooking, and the kamados go to those temps easily (and have mass, etc...they hold heat for a long time for a "grill" and are very efficient!I mean, 800F is fabulous for a good thick steak, but you don't need that heat for long...
But the essential problem with a kamado / egg for pizza is that at temps over 600, without doing really crazy things, the "floor" of the oven is generally a lot hotter than the dome, which means instantly burned on the bottom and undercooked up top
Also, when you open the dome, you lose all your superheated air
So I know folks who try to get the pie up into the dome as much as possible with bricks and such, and yep--they get a 2 minute pie and it looks great--without opening the lid. But fire management is incredibly tricky at this point--hot spots will happen--and you're going to ruin a lot of pizzas on the way to getting it right.
Now, after some thought, I actually think a better way to do a higher temp pie, with a Primo oval, (and maybe *only* with a Primo oval because of its shape), is to build a half fire box fire (on the left, say) and then put two ceramic plates on the right (for building thermal mass).
Then heat the oven to dome temp 750-800 and hold it there for 40 minutes. Then put that Emile Henry--or maybe even a steel plate--on the *right* hand side, and cook a simple pizza on that for less than 3 minutes without opening the lid. Maybe with an extra "lift" in there for a little more height into the dome. But you can't 1) see what your pizza is doing and 2) you can't "dome" the pizza to get a flash of char on top--too much happening too quickly and every time you open the oven you lose a lot of dome heat.
Anyway, that's the experiment I'm going to make. We'll see how it works. In the meantime, I'm looking into wood ovens as well. I do think after using the kamado a lot over a year now that they are better suited (and ideally suited in terms of speed, economy and results) for temps from very low (even cold smoking) to normal-high oven temps (with the ability to get a real sear as well).
Here's a 550 degree dome temp margherita (too low for the 00 flour, but tasted great. It's 60/40 "00" flour with 65% hydration (and too much dough for the stone LOL); buffalo-milk mozza, san marzano tomatoes, salt, pepper, pinch of oregano and fresh basil sitting on the Primo. Absolutely delicious--and 9 minutes total, but not a high-temp pie...