If you look at this diagram though, it seems to me as if it would be much simpler to further heat the steam at the ambient pressure of the autoclave room. Does the pressure of the chamber play some role that I am unaware of? Thanks in advance for clearing this up!
An autoclave can sterilize both solids and liquids, whereas an oven with no pressure control is typically not suitable to sterilize liquids.
Not only do you need to heat the chamber to 121°C, but you also need to make sure that the things you are trying to sterilize are not degraded by the treatment.
For dry objects (e.g. glassware) you could heat the chamber up to 121°C at ambient pressure. In that case, the instrument you're using is called a "dry heat sterilizer" rather than an "autoclave" (REF)
So why use an autoclave at all, and not just a dry heat sterilizer? Quite often the things you would put in an autoclave are water-based solutions (for example, LB broth or buffers). If you do not increase the pressure and try to heat to 121°C, the liquid in your bottles will first boil around 100°C until all the aqueous solvent is in the gas state (i.e. your solutions are going to evaporate) and then the temperature will increase to 121°C. However if the pressure is raised to stay at the liquid-gas phase transition, then the gas and the liquid are in equilibrium, therefore you neither lose or gain any volume in your solution.
So in theory, if you autoclave 1L of medium, then you get 1L back at the end. Note that in real life, the temperature and pressure of the autoclave are set based on the diagram for water, but for concentrated solutions these settings may be slightly off, meaning that the volume in the bottle at the end of the cycle is sometimes a little different from what you put in there at the beginning. See this illustrative diagram of how a solution behaves differently from the pure solvent:
(Copyright: Lumen Learning)
it would be much simpler to further heat the steam at the ambient pressure of the autoclave room
You cannot do that, physically. If you put water in a closed (water-vapor-tight) container, then heat it, the vapor "wants" to expand with growing temperature. But as it cannot expand your metal container (autoclave), the pressure goes up instead. It happens automatically, that's how molecules function in a gas (don't ask me to remember the name of that law please).
You could, I guess, try constructing an elastic autoclave where the pressure stays the same and the volume changes. But that would be mechanically challenging - what material are you going to use that is elastic, won't tire with time, won't deform on you during use, won't burn at the heat source, will conduct warmth well, etc.? Also, you will lose your convenient way of ensuring that you are at a target temperature. Since a certain pressure always equals a certain temperature (at constant volume), you can construct a weighted valve to keep a constant pressure in your autoclave - this is cheap, easily doable with 19th century tech, and more accurate and failsafe than creating a cybernetic control loop with a measuring device such as a thermometer or a manometer.
I don't know if there are also biological reasons to prefer it with with a higher pressure - the number of bacterial species which can withstand high temperature is higher or equal to the number of species which can withstand high temperature combined with high pressure - but I doubt that this was a driving factor in the design and adoption of autoclaves.
The standard autoclave is easy to invent, does its job, does it well, and is cheap and unfussy (when compared to a hypothetical room-pressure-wet-autoclave). The pressure is more of a side effect nobody sees the need to remove - on this site, I'm tempted to call it a spandrel.
As you say it is possible to increase the temperature of the steam over the saturation pressure. And with overheated steam, the instrument would reach the desired temperature as is the case in an autoclave.
But the difference is how many time it would take to heat the sterilized instruments. And the difference is actually huge, we are talking of order of magnitude. The reason is that the heat transfer inside an autoclave is evaporation-condensation. But if the autoclave were not pressurized, there would be a first efficient phase of evaporation-condensation followed by an extremely slow natural convection phase.
Let's try to explain it in few lines.
On the difference between the heat transfer by evaporation-condensation and heat transfer by condenstation
Evaporation - condensation is the most effective way to transfer heat. Just consider the fact that uranium is used in nuclear power plant to boil water.
Transfer of heat by natural convection of gas is very slow.
The difference is many order of magnitude, heat transfer by evaporation is typically of the order of 1'000'000 W/m² (1MW/m²). Heat transfer by natural convection it is of the order of 100 W/m².
What is happening when the autocalve steam is saturated.
Liquid water inside the autoclave boils. Its temperature is the saturation temperature that is a function of the autoclave pressure. The vapor moves and condenses on the surface of the instruments that are being sterilized. The temperature of the liquid film that is formed by condensation on the instrument surface is the boiling temperature. The heat transferred by this evaporation-condensation is huge, in few seconds the surface of the sterilized instruments get close to the saturation temperature and in few minute instruments are homogeneously heated.
What would happen if the steam was overheated without pressurization.
Liquid water boils at 100°C and then a heater in the gaz phase overheat this vapor. The steam reach the instrument and condense. But the temperature of the liquid film that is formed on the instrument is still the saturation temperature: 100°C, not more. So in few seconds the instrument surface reaches 100°C. After a few minutes, the instrument reach an homogeneous temperature of 100°C. Then start a second and slow phase: the condensed liquid films dries, then the instrument get heated by the overheated steam through natural convection, and this is very slow. It would take hours for the instrument to reach the steam temperature.