If you take away the bunkers, essentially what is left is the topography and the greens.
Royal Ashdown Forest works well because of its severe topography.
Similarly, I reckon Yale would be fine if its bunkers were removed, given its awesome combo of topo/greens and Pine Valley would too given its greens and the scrubby areas left behind. Same for Pebble thanks to its topo and tiny pitched greens.
Despite its greens (!), I don't think Winged Foot West's topo is good enough for a course sans bunkers.
Cypress Point would be particularly hard hit by the removal of its artistic bunkers - its greens aren't quite enough.
Interesting to just go the list of famous courses and envisage it.
Cheers,