Understanding thermolysis vs. combustion

Could the heat produced while burning hydrogen, split the very water molecules formed in that process?

No. If you roll boulders down a hill, they release a lot of energy. But you wouldn't expect that energy to be able to fling the same boulders up and over a higher mountain.

The energy released in burning is the energy difference from the initial state of $H_2$ and $O_2$ and the final state of $CO_2$ and $H_2O$. There will be a few molecules that by chance are in a much higher state, but they will be very few and that state will not last long. You would need much more energy to actually ionize/thermolyze the materials.

When really hot water molecules split, what keeps that hydrogen and oxygen from burning up?

The products of burning are low-energy states. Under normal circumstances these states are stable because there's no energy to move them away from that state.

But as you raise the temperature, it is more likely that the energy necessary to "unburn" these products will be available. So at very high temperatures a hydrogen and oxygen atom might combine temporarily and release energy. But very rapidly energy elsewhere from the system will arrive and break the bond. The products favored at low temperatures are not those favored at high temperatures.

Thermolysis

Any substance, when sufficiently heated, will turn into a plasma. In a plasma the substance is atomized, and in addition there is a level of ionization.

For simplicity let's narrow down the substance to a single element. The element is heated up to a temperature where it is a plasma.

In the case of for example Sodium: Sodium has one electron that is far less strongly bound than the other electrons of a Sodium atom. So there will be a temperature where a very large percentage of the Sodium population has lost that outer electron, but not more electrons. As you increase temperature you reach subsequent levels where the atoms transit to ever higher levels of ionization.

At each temperature there is a statistical equilibrium. There will be a rate of Sodium ions and electrons recombining, but there is a significant probability of a severe collision that will knock that electron off again. The equilibrium is an equilibrium of rate of recombination and rate of elctrons being knocked off atoms.

This generalizes of course to all elements in a plasma state.

If you have a plasma with Oxygen atoms and Hydrogen atoms there will be transient molecule formation. There will be a probability of Oxygen molecule formation, and a probability of Oxygen molecules being knocked apart again. Etc.

The higher the temperature, the higher the probability of being knocked apart again. So: the higher the temperature, the smaller the population of (transient) molecules.

Combustion

Generally when there is a setup to combust Hydrogen and Oxygen the setup is designed to remove heat continuously, for the purpose of using the energy of that heat. The obvious example: a rocket propulsion system that uses Hydrogen and Oxygen as propellant.

However, as a thought experiment: what would happen if you combust Hydrogen and Oxygen without allowing any of the heat to escape.

So the setup starts with a mix of Hydrogen and Oxygen, in the perfect ratio, and then combustion is triggered.

With that scenario I expect very hot water vapor, but not a plasma. The very reason that so much heat is released is that the bound state of two Hydrogen atoms to an Oxygen atom is a strongly bound state.

So I expect that the temperature that is high enough to atomize water molecules will be a temperature that is above the temperature that results from the prior combustion.