Joining Paths at Intersections?

I don't have Illustrator on this laptop, so will demonstrate a common straightforward approach in Inkscape. The principle is the same in most mainstream general purpose drawing programs:

(In Illustrator, that would be Object>CompoundPath>Make.)

{Type>Convert to paths.)

(Object>Path>Offset Path. Ungroup. Remove the inner offset path.)

(Pathfinder palette: Subtract button.)

After that, if necessary to individually set the fills, strokes, or other attributes of the individual rings, you can Expand or Ungroup the Compound Path. (In Illustrator, the paths resulting after the punch operation will likely be a Group of simple paths. Other programs may retain the compound.)

The diagonal line to the left of the A in the first image...(which I need to extend to the outermost concentric circle) and the horizontal line below the A are where I want the concentric circles to end.

Those two lines then, are part of what you simply need to "punch" (Boolean subtract) from the rings. Forget the idea about "joining all paths at their intersections." That, as stated, is not possible in Illustrator, and would be an unnecessarily convoluted approach anyway.

I've googled...there is no such feature but is from 2012-2013. Certainly by now, Adobe has added the feature.

The way you phrase it ("joining all paths at their intersections") is not actually what you are describing. What you've found described as not possible is actually joining (making into a single path) more that two path segments at a single anchor point. (Variously described as "making a T join", "joining a path to the middle of another path", etc.). No, that has not been added to Illustrator, nor should it. It's a common question discussed in this forum in multiple threads. There is nothing technically "impossible" about a Bezier curve-based drawing program having an interface which creates such constructs. A few have (Flash's original path drawing interface being the most familiar example). But things like selection routines and path constructs are foundational to a program's interface. They establish principles which behavior details of the rest of the interface build upon. Making such a change in a long-existing program would create ambiguities which would cascade through the rest of the program's established interface, conceptually clashing with other commands and features.

I have played with pathfinder but I'm not sure what happens because half of the image disappears.

Using a Pathfinder operation is probably what you recall doing in the past. There are things you can do with Pathfinder and other path operations which will not "join all paths at their intersections", but will effectively break paths at their intersections, or create additional anchor points at the intersections. But again, creating additional anchor points and then tediously deleting unwanted segments is not necessary for what you are trying to do in this case. As shown, the whole thing can be accomplished with an offset path and a single "punch" operation.

I am to the point where I'm wondering whether its easier just to take a screenshot...trace image feature...fragments I can then just delete...

Avoid auto-tracing. With few exceptions, resorting to auto-tracing is an amateurish practice which effectively negates the advantages of vector-based illustration. You're much better off working to understand the basics of deliberate, efficient, straightforward path drawing and predictable path operations such as basic Boolean combinations (called Pathfinders in AI, but are common in most decent drawing programs).

Features like Illustrator's LivePaint (generically called a flood fill tool) and ShapeBuilder, mentioned above, are standard Boolean operations "re-packaged" as tool-based interfaces and so-called "live" editing. And that's fine; I'm not knocking them. But my advice is for beginners to work with the basic Boolean operations (add, subtract, intersect, difference, etc.) in order to gain an intuitive understanding of the underlying principles which drive such functions in all programs. They are conceptually the same principles you were taught as "set theory" in math classes.

JET

Looks like this might be a use case for the live paint tool or the shapebuilder tool.