As is explained in those links I put up, when you talk about oils you have to distinguish between fixed oils and essential oils. The former includes fats like sunflower oil. The latter being very potent, distinct tasting and smelling natural oils that are used as "building blocks" in many flavourings. Flavourings that are "oil based" (meaning the flavouring is dissolved in a base of fixed oil) should never, ever be used in e-liquid. I'm not sure there is any reason to avoid flavourings that contain essential oils in e-liquid though. I think that would severely limit your choices as so many flavourings do contain essential oils.
Using "oil based" flavourings in e-cigs is bad because the oil base does not readily vaporise. It will just burn in the atomizer.
Exactly! There's a misunderstanding that essential oils are lipids or fats. They are not (generally). The term "oil" is often used for anything that does not readily mix with water, regardless of it's chemical make-up. Essential oils don't mix with water because they are "hydrophobic" (seriously) liquids. They're made up mostly of alcohols, esters, terpens and ketones, and are best diluted in solvents like vegetable oils, PG or ethanol.
I'm not saying all essential oils are safe - they are not. No more than all plants are safe. And even some "safe" plants and essential oils can be dangerous for some people. Rosemary EO can encourage seizers in epileptics (though it's rare), and certain nuts can break someone else out in hives.
But consider for a minute that the preferred method of aromatherapy is inhalation - direct is OK, but the most effective method is with a nebulizer (which breaks the liquid into microscopic particles to breath in via a mist). This is absolutely and perfectly safe. Aromatherapists and people who work with essential oils express so much caution because...you guessed it - the FDA. The UK is SO far ahead of us in recoginizing EOs for what they are - most of the rest of the world is too, for that matter - but that's a whole 'nother can of worms and I won't go there (now, anyway).
The difference inhaling EOs and vaping is that the liquid is not heated. Heating at high temperatures is known to destroy therapeutic effects, but not the aromatic. Whether some substances actually become toxic, I don't know, but I do know that people have been heating natural EOs for scent for centuries. I have no doubt essential oils are safe to inhale and provide numerous benefits, and there is no oily substance that collects in your lungs.
All that said, people having only casual knowledge about them should not be experimenting. Things that need to be considered are extraction methods, grade (pharmaceutical or industry), irritants, counteractions (they are nature's original drugs, and they can conflict with conditions and synthetic drugs, although it's rare), toxicity (sassafras, bitter almond, cinnamon bark [versus cinnamon leaf] among others can to toxic), dilution amounts and methods, etc.
A side note (for whichever side your on), most flavors are synthetic, in that you can't get them from essential oils because essential oils can't be extracted from them (economically or otherwise). Nearly all fruits, excluding citruses, are synthetic. So these are lab-made molecules bursting with artificial flavor and colors (though there may be essential oils from other plants in them as well). Personally, I trust what's found in nature more than I do anything made in a lab, but that's not the world we live in.