Is it good to have the "floating oil" on the surface of some coffee drinks? Coffees prepared with some standard home machines (e.g., non-paper-filtered methods like moka, Turkish, or even some espresso machines) have something on the surface. It looks like oil from the beans.

Filter-based brewing techniques like drip or Aeropress do not result in this oil film on the surface, in my observation. The filter is probably the reason, as it can block the oil and just let water get through.

This oil, if confirmed, is what contains most flavour compounds, so I guess we want to keep it. On the other hand, it is aesthetically nicer to have a clean surface for the coffee drink, which means to act such that this oil is excluded at brewing time.

I am not sure there is a tradeoff here, as "clean" surface coffees are very good (if prepared well), which means the flavour compounds did get down to the drink. So perhaps the surface oil is just that does not participate to the flavour. Any information on this topic?
Nice question. A similar myth has been arisen for Turkish coffee pot; such as it should be cleaned only with water to keep the greasy surface made with coffee oil.

Partly correct, partly not. According to my "Barista's manual" from Lavazza, this sedimented oil may acetify. Thus, end up bad flavor.

Again, (I cannot remember the source) as of my knowledge, paper filters are very good filtering everything; including those valuable oils. (That's why classical French press is still very common to prepare filter coffee, I think.) Paper soak in most of the flavor itself if it does not have proper fibers. This seems like the main problem of any drip coffee machine, V60, Chemex, etc.

Final words: keep it clean in the long run. You may not necessarily clean it until acetifying; which is you may keep it as it is up to one-two days.

Same rule for Turkish coffee pots. :)