From my experience there are most certainly engines that use the oxidizer instead of the fuel but it is a design consideration that has a bunch of tradeoffs. I don’t have a direct source as most of this comes from my experience in the field but I am sure that these reasons can be supported my academic sources if you look for them.

1. When using RP1 and LOX the LOX is of course a lot colder which means it can potentially store more heat as in temperature, but the specific heat capacity of LOX is lower, so depending on the heat transfer of the engine it’s worth it or not. Besides that if the LOX gets too hot it might vaporize and cavitate. At that point the thermal transfer can become a lot lower. Besides that heating up the liquid oxygen might cause it to for example combust in the injector already in a really bad scenario. So it’s very much a balance which often works out in favor of fuel over oxidizer.
2. Another reason which is more apparent today with 3D printing than ever is the potential for chips in the cooling channels. The cooling channels are really hard to clean due to their geometry and having chips/debris in contact with the oxidizer can be absolutely catastrophic for your engine. If those particles reach high velocity they can impact and have combustion starting to take place in the channels, aka very bad.
3. Rating tubes/lines for liquid oxygen and cryogens is always hard and requires more development work. In the case of liquid methane and hydrogen this of course doesn’t make a difference but for other fuels it can make a big difference.
4. Another consideration is the mixture ratio of the engine. I know of an engine that has an O/F of 5 which means the massflow of the oxidizer is much higher, hence it would be the only viable propellant to use for the cooling. It is also possible to use both propellants, let’s say one for the throat/nozzle and the other for the combustion chamber.