...you give the definition of what we call scrub radius...
Yes, that's what I was saying. The only thing that affects scrub radius is where that line intersects the ground relative to the contact patch - NOT the angle at which it intersects. Scrub radius can be 0 with 1° SAI, or with 5°.
...extended to infinity...
I was taught only to the kingpin/BJ axis. On the other side of the front axle, the angle is mirrored
(RONG!!!!); IOW, if the steering linkage is rear of the axle, the TRE at the knuckle will be rear, and lie on that line (to the center of the non-steering axle). But if the steering linkage is forward, it's set up on exactly the same angles as if the non-steering axle was the same WB forward.
But I don't remember actually checking, so now you've got me wondering... If what you're saying is true, you could have both the short rear drag link & the long front drag link connected to both knuckles, and they'd move exactly as freely as if only either one was connected. Intuitively, it seems that having a different-length drag link (or whatever you want to call the one from one knuckle to the other) on the other side would result in VERY-different motion of the steered knuckles; so having both connected would cause binding (inability for the knuckles to steer). But having a mirror-image of the shorter rear drag link attached exactly the same way on the front would allow steering because they should move identically (always parallel).
I'm going to draw it out & check...
Neither of my tests came out right - either because I drew them both wrong, or because we're both wrong.
(phone app link)
In the main drawing & the center inset, I started with the tires on a 92" wheelbase & 57" track width (early Bronco typical dimensions), and added the big Ackermann "V" to their centers (contact patches). The short line from the contact patch inward represents the steering axis, but I had to guess how far inboard of the tire the upper BJ is, based on looking at the '74 I have outside. I drew the circle representing a rear-facing steering arm's TRE arc, centered about as far inboard as it appeared on that truck. The light blue line is a copy of the Ackermann angle, passing through the center of, and extended to both sides of the TRE circle (ntsqd's description of the geometry), representing the steering knuckle's arms. The blue line is copied FOR THE UPPER (driver's) TIRE ONLY in gray, rotated 10 degrees at a time. The first red line is a rear-axle drag link connecting the rears of the original blue lines; magenta is a forward-axle drag link.
Each got copied to the ends of the driver's gray lines; then I struck an arc showing how the drag links would rotate to follow the passenger (lower) TRE circle, and rotated them to meet it. Last, I added the passenger's gray lines joining the ends of the drag links. The fact that they don't remain centered on the TRE circle says that they don't steer the knuckles the same way, so I either misinterpreted some step of the process, or I just goofed the drawing somehow.
Then I erased everything back to the blue lines, and re-drew the forward section of the driver's as a mirror-image of the rearward one (my understanding of the geometry), and repeated the other steps. Except for the passenger blue lines, I copied & rotated them in gray to follow the passenger end of the red drag links, and only inset that close-up at the bottom of the pic. Their forward ends don't follow the magenta drag links, so that version didn't work, either. :-[
BTW
When I was looking at the '74 to gauge the BJ & outer TRE positions, I noticed that the line from the TRE horizontally through the steering axis doesn't come anywhere close to the center of the rear axle - it's probably 3x the wheelbase back (300% Ackermann?).