Why mix cooking oil with epoxy to repair a toilet drain pipe?
I recently learned something about injection molding. While this is a completely different process, it may be somehow related to this topic. 

When plastics engineers want to produce a flexible or partially flexible part, they choose to make the parts from elastomeres in the past (e.g. rubber or silicone). Unfortunately elastomeres are nearly impossible to produce during an injection molding process. Thus flexible parts were rather expensive.

Some smart engineers tried to mix cooking oil into thermoplastic materials like polyethylene and polypropylene to make it softer. They eventually found out, that the outcome was more like an elastomere, but could be produced by injection molding, which makes it rather cheap.
In fact, if you encounter a cheap plastic part like a toothbrush these days which exhibits a flexible part, it is very likely that the flexible part is basicly the same polymer as the adjacent parts but with cooking oil mixed in.

And because cured epoxy has some thermoplastic properties, I think it may be possible to increase the flexibility of the cured resin by thoroughly mixing in some oil. And flexibiliy is the be-all and end-all of a sealing.

This leaves the question, why oil on a surface is detrimental to the adhesive strength of epoxy, and inside the compound it's not. 

I guess it depends on the degree of dispersion. One will have to mix it really good to prevent the formation of an oil film preventing the epoxy to grip the parts.