Peter Watkins said:
... regardless of whether the galaxy is moving away from us, the observer, or we from it, or each from the other?
You haven't listed all the possibilities, or even the most interesting of them, Peter
In GR, distances between remote objects can increase without either of the objects moving.
As far as we know, there is no "outward". There is no outside. There is only expansion.
Mathman already pointed this out.
Peter Watkins said:
Given that the Earth is not at the centre of things, why did several different teams, over 3-4 decades, attempt to "rewind" the [expansion] in order to calculate our age.
!

You amaze me Peter. Why shouldn't they? Expansion has a finite history (governed by the Friedmann equations) so reconstructing the history is one good way to get a handle on the finite age. No centerpoint is required to do this analysis.
This would only work if we were at the centre, wouldn't it?
Come on now! Of course not!
As you know, over this period several different figures were arrived at resulting in the "Hubble wars". N.A.S.A. called a halt to these "unseemly" disagreements in order not to alienate the public, (their paymasters).
CONSPIRACY! Dear Peter it would take more than a word from NASA to make Astronomers all over the world agree and shut up. Getting cosmologists to agree is like herding cats. And the top ones are all over the world. Lineweaver in Oz, Ellis in Capetown, White at the Max Planck Institute Garching, Maartens in the UK.
You shouldn't call it the Hubble
Constant because it is known to be decreasing. People now tend to call it the "Hubble parameter" or the "Hubble rate". I haven't called it the Hubble Constant for several years. If you wouldn't mind, how about joining me in calling it the Hubble rate?
Who then, finally stated what the[current] value of [the] Hubble [rate] is, and given that the rate of separation is increasing, isn't it now redundant?
Wendy Freedman's team was given the job, using the Hubble Space Telescope. They published the current figure around 1998. The evidence was convincing so everybody adopted their figure of around 71. Since then they keep refining the estimate as new data comes in. Narrowing the errorbar and all that. A lot of new data has come in over the past 10 years, but the estimate has changed very little. Just the uncertainty reduced some.
The Hubble rate is decreasing is expected to continue decreasing indefinitely, approaching an asymptotic value somewhere around 60. This is a consequence of the accelerated expansion model in general use, the LambdaCDM.
The rate of decrease is very slow. Won't see a detectable change for thousands of years.
Is the current value of around 71 redundant?
By no means! it is an extremely useful parameter!