I think this is a very interesting subject. Some modern biz jets rely exclusively or almost exclusively on GPS for FMS position but also other key systems. I do vaguely remember an incident a few years ago on a Phenom 300 where the loss of GPS did indeed lead to the failure of other major systems such as AP/YD and possibly even AHRS. I remember thinking at the time: "that's not good..." Maybe a current Phenom 300 driver or one with better memory than I have can chime in.
I have had similar experiences to the OP on a Citation using G5000. Within moments of losing both GPS we had gone from normal ops to DR. DME/DME did not provide an acceptable FMS position. It doesn't sound like a big deal but a lot of stuff was affected: No synthetic vision, incorrect aircraft position on map. Incorrect wx radar picture with reference to the map, GS wrong, wind info wrong. Distance and time to go wrong, fuel at destination wrong, displayed MSA unreliable. So a lot of the info we rely on to form our SA on a day to day basis was no longer available or unreliable. And of course these things happen mostly in places where there is either no radar (such as Cyprus) or where the work load is already quite high (such as Moscow).
I will be interested to read the experience of others.