LOL no. Modern full suspension bikes even with a lot of travel can climb great. I just demoed a Pivot Firebird with 170mm of travel and the thing climbed nearly as good as my Surly Karate Monkey rigid and it weighed less than 30lbs, also cost over $6k lol. They have done a lot with modern geometry, rear suspensions, and shocks to make them climb well. My main bike is a Intense Spider with a 140mm front and 130mm rear and again it climbs nearly as well as that Surly, climbs better than my on-one hardtail. Now a full on downhill bike isn't going to climb for crap but nobody would expect it too. Oh and MOST people that purchase mtn bikes and know nothing about geometry are noobs buying your standard 100mm XC hardtails that have the same old steep head angle geometry that has been around forever and again climb without any problems.

As for finding a rigid it isn't that hard. Most people want at least a suspension fork so you have to look a bit and most are more niche brands but they are out there. Specialized even sells one, think it is a version of the Fuse? I ride with somebody that has one all the time setup single speed and it weighs 20lbs. It is nothing special or expensive, just an aluminum frame and stock everything. Half the people I ride with are on some rigid single speed bike.

As for shuttling people do it because they can and it gets them more downhill runs in a shorter amount of time duh. Downhill is more fun than uphill at least to most people. Seth's Bike Hacks over on youtube just did a video on this lol.