"Would it have made a difference if I'd said I am a sibling of one of the aforementioned parties and did not approve of this course of action and was not happy to be asked to be an intermediary?"
A sibling, who is involved with both parties, is very different to someone who is "merely a spectator", so why didn't you say so?
(In fact, I didn't really buy the "merely a spectator" role, but thought you might well be the father).
So is that the actual point of your OP?
That the mother "does not want to see him (the father) even for a brief handover of the DCs and she wants respective family members on each side to do this" and that you, a sibling, have been asked to be one of those respective family member, but that you do "not approve of this course of action and (are) not happy to be asked to be an intermediary"?
Is that it? If so, then you should refuse. You should not become involved with the handovers if you are not comfortable with doing so and, as part of the point of third party handovers is to make things easier/pleasanter for the children, you would not be the best person to do this if feeling uncomfortable.
With regard to you feeling responses on this thread have been unfair or unfriendly, I do think you rather set yourself up for that, by misrepresenting your involvement (if you see what I mean).
As a mere spectator, you would have deserved to be told to butt out of what is bound to be an unhappy situation for all those closely involved.
As a family member who has been asked to facilitate handovers, it's fair that you should be able to express your discomfort (or even disagreement) with the situation (although my advice remains that you should simply refuse to be involved and leave the parents to arrange third party handovers in some other way).