You're quite right, but my point is that as far as the GPS is concerned, an approach is just a sequence of waypoints. Whether those form part of a published approached, or whether they're user-defined ones that you made up as part of your home-brew approach, it neither knows nor cares.
Actually the GPS does know and care, very much.
With a published approach loaded the GPS will scale at the FAF from 1 mile to 0.3 mile full scale deflection and a TSO 146 GPS will scale to full scale deflection of 2 degrees inside the final fix down to 350 feet at the missed approach point.
A homebrew approach cobbled together using just a sequence of waypoints will be scaled to 1 mile at best and could be as much as 5 miles full scale.
I know what I'd want to use for an approach and it sure ain't some dodgy homebrew one.