This was going to be an email until I remembered: comments!
I’ve always understood “fast approaching” as reflecting our subjective perception of the passage of time: future events seem a long ways off for a long time, then suddenly they’re about to happen and we don’t have enough time to prepare for them. Cases in point: paper deadlines, final exams, packing for a move, etc.
Seen in this light, the time-per-unit-of-time problem is less troubling. The relevant measure for time’s “velocity” becomes “the amount of time that has passed (or that I observed passing) per ‘beat’ of subjective attention,” where the “beat” refers to a particular moment of paying attention to something. This measure also explains why a watched pot seems to take longer to boil: the amount of time it actually takes is the same, but the more I direct my attention to it (the “beat”), the longer that amount of time seems to take.
(That loosey-goosey formula should probably be revised to take into account the well-known phenomenon that something pleasant seems to pass quickly, even if you’re giving it 100% of your attention, while something painful that also commands 100% of your attention seems to pass slowly.)
I’ve also always thought that “fast approaching” meant that some future event was approaching us. That seems to be how people use the intransitive form of the verb: e.g., “the train is approaching” means the train is going toward some object; “I’m approaching the base now” means that the speaker is going toward the base. I agree that “approach,” even used intransitively, has an implied object, but that property is no different from other verbs (e.g., “I’m leaving [my house] now”; “I depart [from home/to work] in the morning”; even “I read [stuff] in the mornings”; and more slangy usages like “I lift [weights] in the gym, but I also do aerobics”).