I don’t understand the purpose of applying oil to the pad when French polishing.
It's a lubricant.
That's it basically. Although there have been deliberate attempts to protect the process and keep it under wraps in the past1 there's no jealously guarded proprietary knowledge protected by a Mason-like secret handshake here ^_^ the oil is just to lubricate the pad and prevent it sticking.
Some sources say to apply it to the bottom of the pad - wouldn’t that interfere with the shellac?
No. I know it seems like it has to, but just be confident that it doesn't. French polishing has been done this way continually since the 19th century which tells us it can't be a problem.
Others say to apply it to the sides
I was going to say I've never read this but TBH I just don't recall. I do believe the commonest advice is to oil the pad face, either directly or by dipping a finger into oil and tapping that to the face (depending to some extent on how much oil the specific source you're reading suggests is the right amount to use2) and I can confirm this works exactly the way you want it to.
1 Read with this in mind, older sources in particular.
2 How much oil to use, how exactly it should be used and what oil you can/should use, you have to delve more deeply to find out.... or figure it out for yourself. Trust me, it's not that hard. Once you realise the oil forms no part of the final finish and is only a lubricant you'll probably realise the same thing that a number of modern practitioners have independently figured out, or found out via experimentation.