'If Archbishop and Chancellor have equal value, it DOES NOT IMPLY ANYTHING
for the value difference of Rook vs Bishop. They are all different pieces,
and have nothing to do with each other.'

YES it does according to my model and every quality, holistic model built upon a proper foundation I have ever seen.  Contrary to your statement, I think it obvious to any logical person that the component pieces have at least SOMETHING to do with their composite pieces.

Some computer chess programmers are notorious for achieving useful relative
piece values that are within decent range of their optimums based purely upon AI playing strength without creating any coherent, fully-developed theory that is logically explained, justified and consistent.  Unfortunately, such people contribute little to the understanding of
relative piece values for themselves or other interested parties.

I have appr. two years of experience working with Reinhard Scharnagl's 
excellent SMIRF program, my fast dual-CPU server and choice Capablanca chess variants.

Reinhard Scharnagl would compiled two, otherwise-identical versions of his 
program using his and my favorite sets of relative piece values (at that time) which would played against one another using a great amount of time per move.  Eventually, we carefully completed many games this way.  We would both analyze the game results and discuss conclusions.  Sometimes we would agree.  Sometimes we would disagree.  Subsequent tests would settle disagreements ... sometimes.  In this manner, we both improved our models over time until we reached a point where any further minor improvements became prohibitively difficult to achieve within a survivable time frame.

'In real life the value of a piece is not the sum of the value of each of its individual moves ...'

YES it is although not exactly.  The moves of component pieces of a composite piece have far more effect upon determining its relative piece value than ALL other factors added together.

'... but also depends critically on properties like mating potential,
color-boundedness, forwardness, speed, manoeuvrability, concentration,
sensitivity to blocking.'

I have also read ALL of the pioneering works of Betza on the subject.
Essentially, my model mathematicized a subject (to the extent possible
present day) he had only speculatively verbalized.  Rest assured, my model 
makes quantitative adjustments for all non-trivial, effecacious factors to relative piece values that I know of with certainty.

You need to read and thoroughly understand my 58-page paper on the subject.

universal calculation of piece values
http://www.symmetryperfect.com/shots/calc.pdf

'Theoretical considerations like you refer to are just nonsense, with no
connection to real life.'

The connection of my model to 'real life' is very strong.  My theory was
adjusted and refined numerous times to comply with game results over
different piece sets and game boards.  Experience dictated the details
of the theory in accordance with the scientific method.

'What would you rather have (if you can choose to make a trade or
not), a piece that is more 'valuable' according to some contrived
reasoning, or a piece that gives you a larger probability to win the game?'

Both.

Under a proper model, they should not be mutually exclusive at all.
In fact, they should be in agreement ... until a point in the endgame
where checkmate becomes possible.  Be mindful that significant
differences in relative piece values between the opening game, 
mid-game and endgame (to the limited extent that they are applicable)
are accommodated under sophisticated models.
____________________________________________

See the published values of Ed Trice and Reinhard Scharnagl for
CRC pieces upon the 10 x 8 board.

http://www.gothicchess.com/piece_values.html

In addition to the published values of Hans Aberg and Derek Nalls, this verifies that it is beyond dispute that your published values for the archbishop and chancellor are radical.  Your radical contention that an archbishop and a chancellor have appr. equal relative piece values requires an especially sound theoretical framework to be convincing.  Instead, all I am receiving from you is piecemeal descriptions of endgame scenarios where material values are likely to disappear and become meaningless compared to positional values (i.e., checkmate achievable regardless of material sacrifices) and consequently, conclusions drawn are likely to be faulty.