A wood-fired oven that works properly will hold cooking heat for hours after the fire dies down, bake a pizza in ninety seconds, and still be warm enough for bread the next morning. One that does not will burn through a pile of wood, never quite get there, and cool off the moment you stop feeding it. The difference is almost never the brickwork you can see.
An oven is a heat battery. The firebrick dome and hearth soak up energy during the fire and give it back slowly while you cook. That only works if the heat cannot escape downward into the slab or outward into the night air, which is what the insulation layers are for: an insulating board or vermiculite layer under the hearth, and a ceramic blanket or insulating render over the dome before the decorative stone or brick goes on.
Those layers are the first thing cut from a cheap build, because they add cost and are completely invisible in the finished photo. They are also the only part that genuinely cannot be added later — reaching them means taking the oven apart. Everything else on an oven can be repaired, refaced or extended. Not that.
The other thing worth getting right before you build is the flue. It belongs in front of the dome opening, not on top of the dome, and it needs enough height and the right throat proportion to pull smoke up and out. Get it wrong and every fire vents across the oven mouth and into the face of whoever is cooking, which is a miserable way to discover the problem.
Under all of it sits a poured footing and a stand rated for the weight, because a finished masonry oven is heavy — a few thousand pounds is normal — and anything that settles under it cracks the dome above it.
So when you compare oven quotes, look past the stone. Ask what goes under the hearth, what goes over the dome, how tall the flue is and what the footing is. That is the oven. The facing is the jacket it wears.