A little experimentation suggests that CCleaner is not the problem. It works fine on my other PC (same configuration).
On the "broken" one I restored an earlier version of the Firefox profile, but it seems to be corrupt - the history list does not get updated when I visit a new site.
So I re-created the Firefox profile, re-imported the bookmarks, surfed the web a bit (history list gets written correctly), then re-ran CCleaner, which now behaves correctly - deletes the history and cookies (other than exceptions), but does not delete the bookmarks.
So it's all good now. CCleaner keeps its reputation as robust and useful software!
By the way, the reason I'm using "so old version of Firefox" is that the newer versions don't have any features that I want - only bells, whistles and bloat that I don't. That seems to be the way with an increasing number of software packages these days - the newer versions are not necessarily better, and sometimes are worse, than the older versions. CCleaner is one of the few that I'll upgrade regularly.
