In our classic 737s the EGPWS had for the reasons mentioned its own GPS receiver which was deemed more accurate in more situations than the pure dme/dme update. In the NGs that is not really a concern anymore as the usual ANP hovers around 0,02NM. Besides, you can retrofit the classics to use GPS position with the FMC as well, however that does cost money of course.
Didn't they install a new DME near INN now to increase "conventional" RNAV accuracy as backup to GNSS based RNAV? In other parts of austra too as far as i'm aware. Our main problem area back then was russia where there was no DME update available even in the moscow area, now that has changed as well.