but if the DME and GPS agree then you are almost certainly where you think.
Come on you lot; the difference between GPS and DME is under 0.1nm in almost any flying context.
Is GPS distance relative to the threshold of the runway in use or the ARP? Depending on the answer, the airfield and the procedure the DME and GPS distance difference could be several miles. Even the slant distance at the start of a typical base turn you see probably about a 0.5nm difference (I think it should be sin(3), not cos).
And how do you check that the DME is working when you start an approach that needs it?
You will know because you have ID it been using it before you start the procedure and its reading corresponds to the other nav instruments which agree with each other. Many airfields have timed approach procedures, some don't. If not and DME is wonky then you cannot do an instrument approach. Where permitted, it is standard practice to start a timer at the FAF/FAP for all approaches so that it is possible to revert to a timed approach should the DME fail.
would you ever be happy with completly relying on a GPS approach without any other positional cross check in circumstances where a gross error would place you in terrain before becoming visual?
I don't think anybody would be happy with just one instrument. Most approaches involve two or three with the non approach receivers tuned to other navaids (if possible).