Spoiler: by being meticulous in trying things, I got this working, thanks to your confirmation that I should focus on “apply transform offsets”.
Thank you for the quick reply (on a Sunday!). I’ll try to be completely clear with images as to not leave you guessing 
First off, here is my Animator component on the “PlayerFPC” game object (I’m unclear about what that first “Root position” info box means):

This is what the animation window looks like for the clip I’m having trouble with (as you can see, it does have position and rotation, and position is also very broken and its easier to inspect, so lets focus on position):
And this is what I mean by the Transform (mind you I use TransformPro) being empty when I lock the inspector panel to my object being animated (PlayerFPC) and then try to see where it is animating the player to when I am completely in magical somewhere land:

However, I was able to see values when viewing the debug mode inspector (sorry, I didn’t try this last night), but the rotation is given as a quat so lets look at the position instead:

For the record, the base local position of Frame 0 should be (10.53379, 3.47571, -7.478381).
The end of the mystery is at hand:
Now, for all you folks at home, here’s where I made a mistake yesterday and today: I always focused on one of the two values for the offset (I thought positions are obviously broken, so I add a position offset to the clip). But for some stupid reason, I never put both offsets in (position AND rotation).
As soon as I put both offsets in (as collected from the first frame of the old animation), the clip started to work!!!
So this is solved, thank you, thank you very very very much @seant_unity .
I do have one related question
What the heck do these buttons do?

Thank you again.