"Does it not seem reasonable that, of all the diversity of living things, at least some fraction would be due to mutations that really did just plain survive at random? Things that weren't the best adapted of anything, but were viable enough to hang in there?"

It seems reasonable, but it is not how evolution works. If a species has survived it's because it's well adapted to the environment it lives in. Every single living organism, without exception, is subjected to the pressures of natural selection, which is itself not a random process. Every organism is part of an ecosystem, part of the transfer of energy via tropic cascades and other processes that sustain life on the planet.

How certain species persist for a long time is interesting though. These species tend to be generalists and adapt easily when environmental circumstances change; the more specialised an organism is the more vulnerable it is to environmental change. Even the most long-lived species will eventually become extinct though, they might become extinct or speciation might occur as natural selection selects the best adapted individuals.