It's still acidic, but for instance Aeropress says coffee brewed with an Aeropress and paper filter has about 20% the acidity of a regular drip brew (and about 10% of a french press). (As far as I know this is largely down to paper filters and lower brewing temperature, both of which you can emulate other ways.)
Obviously only a personal anecdote, and for most people it'll never make a difference, but as your stomach becomes irritated it doesn't take much to make it worse. On a bad week, just drinking some black tea (ph~5) rather than green (ph~8) or putting a bit of black pepper on something can be enough to have me in pain for hours. Think like, normally it might not hurt to poke your arm, but if you've got a big bruise it doesn't take much to make it hurt.
On a good week, drip coffee through a metal filter is largely fine. On a bad week, it's painful but coffee from the aeropress still sits mostly okay.
I'd be interested to know if reducing that oil actually helps with heartburn though because I'm surprised there'd be enough there to cause problems. Or at least to cause noticably more problems than the caffeine and acidity of the coffee.