The problem with pizza is that generally the toppings/cheese cook at a different rate than the crust, with the toppings/cheese taking longer to finish than the crust. Most pizza ovens are convection style that will have an arrangement of fingers (blowers) above and below the pizza, usually twice as many above as below to ensure an even cook. Basically a majority of the heat is coming from the top down.
A hearthstone oven typically uses radiant heat and will have a heated domed top where the dome will be much hotter than the cooking surface. Because heathstone ovens are typically designed for very fast cooks they are very hot (900F on the cooking surface, up to 1200F in the dome) so they typically only accommodate a very thin crust. When you are running a hearthstone oven you basically put the pizza on the cooking surface until the crust is done then lift it into the dome to finish off the toppings.
The issue with wood-fired is generally the heat comes from the bottom up. So either your crust is done long before your toppings (meaning you burn the crust to cook the toppings) or you have to really really skimp on the the toppings to get them to finish with the crust. Most places I've seen go for the "burn the crust to cook the toppings" method. Even woodfired ovens that accommodate for this fact in their design still have flare-ups that will also add burned spots to your crust. Don't get me wrong, a well designed recipe and crust mix combined with someone with a lot of experience running a wood-fired oven can absolutely make a great pie. However, most places are just using an off-the-shelf dough mix and are hiring high-school/college aged kids who aren't even around long enough to learn the nuances of running a wood-fired oven.
This is why for at home I recommend a pizza stone combined with a convection oven. The stone provides a nice even cooking temperature for your crust but blocks the convection from hitting your crust, allowing your toppings to cook a bit faster and be properly finished by the time your crust is done.