Potentially a much smaller headache than pumping one or two more fluids around inside a jet engine and keeping them from breaking down. The jet fuel only needs to not break down for a single pass through the engine. A closed lube or cooling system would need to not break down for multiple passes (i.e. much longer duration at high temperature).
>it must just be really hard to find anything with the right thermal stability.
A total loss lubrication system doesn't need to be actively cooled (assuming your lubricant input is of suitable temperature) which saves a ton of weight and complexity which is why rocket and jet engines often use the fuel as a lubricant.
The plane had a limited number of TEB "shots" and those were needed after each afterburner use, or when restarting the engine in mid-air (as crew needed to do under some conditions), so they had to be carefully managed.
I guess the main requirement for the latter two purposes is that it doesn't break down at really high temperatures. The jet fuel needs to be stable at high temperatures too (additionally, you need to be able to burn it to make the engine go) so once you put all this research into finding the right jet fuel you have a liquid that will do OK on the other jobs.
Edit: and also it was used for lubrication! Good fuels definitely do not necessarily make good lubricants...it must just be really hard to find anything with the right thermal stability.