I doubt there's any component in a PC getting remotely close to the ignition point of oil (which for canola is 424°C). Plus, it's going to be a minimal amount of oil.

I'm more worried that the oil won't get to a high enough temperature and thus won't polymerize, so it'll flow out and ruin some other component, or go rancid, or something. Thermal paste won't move on you. Oil will.

Oil will certainly move on you, but it might not destroy your components, depending perhaps on the specific oil chosen: you can actually buy or build fully oil-cooled PCs.

https://www.pugetsystems.com/submerged.php as an example.

That's mineral oil, not cooking oil. Mineral oil doesn't go rancid.

Presumably there’s enough surface tension from both sides to hold it between the gaps and resist gravity (if it’s a vertical CPU).

Pretty much any Solid-liquid-solid or solid-solid-solid interface will be better than solid-roomTemp&Pressure gas-solid.

The whole point is to conduct heat better than air, and most things will.