Lee:
I've had a look at your site, and read with interest (looked at pictures) at the work you do.
The only comment I'll make, is that the methods described and shown will make every trained engineer wince.

That doesn't mean you don't do a great job, or aren't a great fellow, but the methods you use do look a bit, well,
primitive.
As far as the remark about 'bodge' were concerned, I was supporting your comment actually. So if you don't
now agree with me......?
Getting back to removing a heel tip from the sprung tube..... A task you believe should only be carried out by rocket scientists, or a cobbler ....
The heel tip is splined. Why? To ensure there isn't full contact with the tube, which is made of a material that (a) oxidises (b)
can produce electrolytic transfer of material. Ideally, heel tip removal is through a direct in-line pull central to the centre line of the tube. If the heel tip is oversize, or has welded itself to the tube, (as above) the tube will be pulled out of the heel. [As happened in my case.] There is tolerance in the 'spline within tube' design for the heel tip to be rotated in a circular motion with the tube centre being the centreline to the rotation. As the spline has edges, continuous single direction movement can cut material, and make the tube large enough to make the tube too large to hold a replacement tip. A way to avoid this, is a see-saw action. Imagine turning the minute hand on a clock from 5 past, to ten past, then back to 5 past again. The technique can alllow the spline to free itself just enough to remove the tip. If it remains 'welded' to the tube (as in my case) minimal amounts of rotation of the tube in the heel shouldn't cause further damage.
If pulling the tip pulls the spring tube out, this isn't a problem, provided the tube can be returned to it's original position and locked back into place. There is an 'opening pin' fitted at the other end of the sprung tube that increases the outside diameter of the tube deep inside the heel.
In my case, I had to use a slightly slimmer pin to knock out the welded-in heel tips. I've cleaned up both tubes and reinserted them with their respective locking pins. If they subsequently show signs of loosening, I'll araldite the base of the tube in place. The value of the shoes isn't so great I would replace the tube, or replace the heel. Replacing the whole shoe would be cheaper, and I'd get the 'new shoe' look included.
That's the story of my boots, now back 'on thread'.
As Lee has already suggested, it might be prudent to show your heels to a quality cobbler. They are hard to find, I can tell you.
Only last year, I had my friends boots re-heeled at one of the local "cobblers". The first had said the tip on her plastic [blade] heel wasn't refurbishable, and wouldn't take on the job. The one in the next town had no such problem. However, the next time the boots were worn, one of the staples used for the re-heeling broke out of the heel. It produced a dangerous and terminal split that thankfully occurred close to home. Timpson
could replace the heels, [at a cost of £18 each or £30 pair] but that cost is more than replacement boots. [The style is not worth much commercially, so are sold very cheaply these days.]
I have this morning asked at a local Timpson what the chances are of them acquiring 5" (or higher) heels. They weren't at all sure, and the shoes would have to be sent away for a week or two, to the company that does their heel replacements to "see what they could do". The inferance was, possibly their supplier might find something suitable, possibly they wouldn't.
So we are back to making good what's already available. As I've said, the heel tip spline seems too small, and that's rather obvious if the tip falls out.

A cobbler will know if it's the tip spline that is under size, and installing the correctly sized heel tip is the easiest/cheapest solution. If the spring tube is over sized (worn or mistreated) it's also replaceable, but at slightly higher cost, obviously.
As Lee has said, both jobs can [and in this case
should] be carried out by a 'competant' cobbler.

And given the probable value of the items needing repair, I too think it prudent you go down that route. As I said before.
..... ...