Besides the safety concerns mentioned by Hobbes, there are other things to consider.
The gases at the combustion chamber are at a very high pressure and temperature and as such are excellent at driving turbopumps. They are exhausted at a much lower temperature and pressure, having given up a considerable amount of useful energy in an efficient expansion cycle. More importantly, the turbopump is able to achieve a higher pressure (albeit at lower volume flow) than the combustion chamber pressure, as it is effectively 2 independent machines connected by a shaft: a turbine and a pump.
In a pressure fed cycle, gas from the combustion chamber, besides being very hot, is necessarily at a slightly lower pressure than tank pressure. It is difficult to make a lower pressure gas pressurise a higher pressure tank. Surprisingly, it is possible if the low pressure gas has greater volume and here are some tricks employed in steam engines to make this happen, but it requires a considerable waste of fluid. https://en.wikipedia.org/wiki/Injector (see section on feedwater injectors.)
The other issue is the nature of the combustion gases. They are mainly H2O and CO2. They need to be cooled before entering the tank, or, if not, they will reach the temperature of the tank contents as soon as they enter it.
In Oxygen or Hydrogen, these substances will instantly freeze and be useless at maintaining pressure, other than by the heating effect they have on the tank medium. In a kerosene tank, the H2O would condense and sink to the bottom, which might not cause problems if the percentage was small. Only the CO2 would go to pressurizing the tank.
All in all, it's more complicated than you make it sound. A separate helium tank is a better solution.