The ICAO and EU countries, as I understand it, have kept the requirement to tune and display the underlying navaid, unless it is a RNAV approach.
The ICAO maybe, the EU has changed things as well.
Of course you can fly it using the old fashioned procedures by tuning the VOR, calculating your rate of descent (drive and dive is not allowed anymore) and flying it in VORLOC using V/S. But that is not necessary anymore of course. And of course we are certified for RNPs down to 0.3 with either GPS or only DME/DME and down to CAT I (currently flying CAT IIIb tests) using GBAS or anything inbetween.