Yes they do care and can get excitable about it (in Mexico they will gently correct you). If your going to Iceland, for example, they will get all on their hind feet if you don't file correctly. In Europe they can be downright twits and charge extra handling charges (remember, just for an everyday normal flight from Athens to Scotland, $790 Euros) if you get it wrong. Yes, it is simply G if your a regular WAAS GPS, S for transponder, if your ADS-B Compliant E. All the other stuff has to do with RVSM separation equipment. Remember, if your GPS isn't WAAS, outside USA, you cannot predicate and file as a G.
A whole lot of the world has no working radar or nav equipment. It is there on the charts, just doesn't work. ICAO will fix stuff, but they are real slow. The locals don't care. Typically it is an electricity supply thing, no fuel for the generator, or the grid is down. Some places will swing the radar antenna for looks, but the systems is long dead. Like Greece, lowest standard of living in Europe (quote from "Time Bandits"). I finally got suspicious when they cleared me to decend into a mountain. I asked, "Decend before or after the mountain?" "Uhhhh, after." So I went over to the tower and looked, everything was dead and the approach control guy was using a plotting board.