Why isn't time just a function of the speed of light being finite
In a way it is. If the speed of light was infinite, everything would happen at once. And it doesn't. But more generally I think it's better to say time is a function of motion. The mechanism of a clock is called a movement. A clock doesn't literally measure the flow of time like it's some kind of cosmic gas meter. A clock "clocks up" some kind of regular cyclical motion and displays a cumulative result called the time.
Time is generally referred to as the fourth dimension
Yes. Like Jim said you need the four coordinates to specify an event.
and seems frequently to be treated as an equivalent to the 3 spatial dimensions.
It isn't the equivalent. There's a minus sign in the t term in the spacetime interval.
But then people seem to get tied in knots as to why time has an arrow, i.e. it progresses in one direction only but we can move freely in the 3 spatial demotions.
They do get tied up in knots with this. IMHO the point they don't appreciate is that the time dimension is a dimension in the sense of measure, not in the sense of freedom of motion. I can hop forward a metre but you can't hop forward a second.
The thing I can't get my head around is why time is anything more than a function of the speed of light, i.e. time is defined by c rather than c being defined as distance/time.
In a way it is. The "coordinate" speed of light varies with gravitational potential, and optical clocks go slower when they're lower. Then we talk about gravitational time dilation. But these clocks don't go slower because time goes slower. There is no actual time flowing inside these clocks. This book about Gödel and Einstein is worth reading. It's heavy going at times, but I liked it. Note though that the blurb is misleading. Time exists like heat exists, and a hundred years will kill you just as surely as a hundred degrees C.
Consider a photon leaving the sun, because it cannot travel to an observer instantly there is a "propagation of change" across space at the rate c.
Yes, we say it takes 8 minutes. But think about what we're really doing: we're comparing the motion of something inside the clock with the motion of the photon through space. And our clock could have been an optical clock.
It would great if someone could explain why time is a dimension and not just a function of the "speed" of light?
I think Jim did that well enough. Check out the word dimension in the online etymology dictionary:
late 14c., "measurement, size," from Latin dimensionem (nominative dimensio) "a measuring," noun of action from past participle stem of dimetri "to measure out," from dis- (see dis-) + metiri "to measure" (see measure). Meaning "any component of a situation" is from 1929. Related: Dimensional; dimensions."
It's to do with measurement. Time is a dimension in the sense of measure. That's what the word meant: "a measuring". Temperature used to thought of as a dimension too, but it isn't any more. The meaning of the word has gradually changed, and now people think of a dimension as something that offers freedom of motion, not something to do with measurement.