A common fuel amongst rockets is that of the combination of cryogenic liquid hydrogen and liquid oxygen. I looked at the design of a simple rocket using this cocktail that is the propellant and found that the aforementioned propellant needs a pump or an inert gas to pump the liquid hydrogen and liquid oxygen into the combustion chamber of the rocket. Cryogenic liquids have extreme expansion ratios when their boiling point is reached and the gas expands within the container, and if it is a sealed container then this means that pressure builds up. My question is thus; can't you heat the liquid hydrogen and liquid oxygen within the fuel tanks of a rocket, the cryogenic hydrogen and oxygen expanding and creating pressure in the sealed fuel tank, and use this pressure to pump the fuel into the combustion chamber of the rocket instead of using a pump or an inert gas?
This is known at autogenous pressurization https://en.wikipedia.org/wiki/Autogenous_pressurization The tanks are pressurized to help maintain structural integrity and to aid in the intake of propellant to the turbopumps. But pressurizing the propellant tanks is not the same as forcing the propellant into the combustion chamber. The exit pressures from rocket turbopumps are very high indeed, higher even than the pressure in the combustion chamber itself.
For example in the Raptor engine the pressure in the turbopumps is believed to be in excess of 600 atmospheres. The combustion chamber being around 300 atmospheres https://commons.wikimedia.org/wiki/File:Raptor_Engine_Unofficial_Combustion_Scheme.svg
The main propellant tanks of Starship (which the Raptor engine is being built for) have been pressure tested and rupture at 8-9 atmospheres, so putting them under 600+ atmospheres would just cause an explosion and even if it were possible it would consume vast quantities of propellant to fill the tanks at 600 atmospheres.
Another issue that may arise with autogenous pressurization is condensation. This is a particular problem if the rocket is intended to make any significant changes of direction while the engines are firing (as Starship was). If sloshing and cryogenic spray occur inside the tanks during a maneuver the hot autogenous gases are in contact with a vastly increased surface area of cold cryogenic liquid and condense rapidly reducing the pressure. This problem was a major contributor to several prototype Starship crashes. I believe SpaceX are now augmenting the autogenous pressurization of Starship with helium at least in the short term.