The Brits were big on touring with fixed gears up until the 1960s or so. Apparently, there were two major cycling factions in Britain: the British League of Racing Cyclists, and the National Cyclist's Union. The former were into Italian style bikes with derailleurs, while the latter were into more traditional "classic" British bikes (Hetchins, etc.) and used the fixed gear for everything. Certainly those who preferred fixed did not seem handicapped by it; I've read stories of folks taking extended tours of hilly country, packing their camping gear, and using only a single fixed gear.
It's also worth noting, though, that most of the touring done in Britain at the time was very lightly loaded touring, with just a large saddlebag, carrying enough supplies to see the rider between hostels or B&Bs. While I have not done any loaded touring with my fixed bike, I have done quite a few 200 kilometer days in the hills with it, and would be perfectly happy doing extended "credit card tours" with just the saddlebag in place.
The gear, BTW, is 65 inches, which I personally find to be the best compromise for long rides in the hills. And yes, I wear rubber-soled touring shoes for walking up the steep bits!
(And sharp eyes will note that by the time this picture was taken I had moved into the true mountains, and started cheating by adding a 24 tooth freewheel to the other side of the hub.)