Can an oil bath replace a heatsink and fan?
Liquid cooling can be, and is used in some electronics equipment.  

You still end up with the requirement to remove the heat (and you did mention using convection and conduction). So you end up having a heat sink (or other heat removal device) somewhere remotely located from the heat source.  

The benefit of liquid cooling really comes into play when the liquid is circulated (moved), moving the hot liquid to the remote heat removal device. Without movement of the liquid, you end up with just a heat reservoir.  

Another consideration to take into account is the purity of the liquid. If contaminants get into the liquid, negative results could occur.  

Also, liquids have different specific heat and thermal conductivity, which would need to be considered.  

Resistivity of the cooling medium could be of concern if your electrical contacts are also submerged.

EDIT 1 :  Caution :  I am not sure that it is relevant here, but flammability of oil should be considered as potential hazard.  

EDIT 2 :  Reference Reading Material on liquid cooling