I am interested in comments on a fix. Sometimes we run into conductors in a ceiling box that have been damaged by overheating due to over wattage of the incandescent lamps. To restore them, is it permitted per Section 300.14 NEC to cut them back to good insulation and splice a six inch tail? If not, what can be done?

I always first try and pull some slack through the clamp or connector if possible. If not, I will do exactly as you described.
In older homes where 60C NM cable is present, I always connect pigtails of 90C NM-B wire to comply with the light fixture listing that they be connected to min 90C wiring.
As far as 300.14 goes, it could probably be argued that there doesn't seem to be any wording that indicates that the required free conductor has to be one continuous piece and since this pigtail conductor doesn't originate from outside the box through a separate KO entry, it doesn't count as any additional box fill according to 314,16[B],1.
So, IMO these pigtailed wires would be considered a single conductor and the wirenuts also wouldn’t need to be counted in box fill either.
Seems like a win, win situation for once.

Would it be better to use spaghetti tubing (or heat shrink, whichever listed for the purpose) to slip over the damaged portion of the insulation on the existing conductor? (remove the crumbled or scotched insulation, and use the spaghetti tubing as new insulation, and overlap the tubing over existing good insulation). That way, there's no splice in the copper itself. You could use spaghetti that has a higher temperature rating.

I have used both of these fixes and butt splices as well.

It sure beats having to cut in new boxes and "invisoplates".

Glad I'm not the only one.

Went out to fix "a few lights not working" last week.
Without wasting too much time I decided to take down the entry fixture. 6 old asphalt cables in a 3/0 metal with a hickey and nice and crispy. Oh boy! Two j-boxes in the attic later everything was working.

For a short while, right after AMP started marketing the enclosed NM splice/tap some were suggesting this could be used and buried in these situations.
Knock out the old box, put in a splice up past the box and reinstall an old work ceiling box. Fortunately that didn't last long. The code 334.40(B) didn't change, only the marketing and literature that comes with the device. Now they only suggest this can be used in modular homes.
It is a distinction without a difference to me. Fires don't know the difference between stick built and factory built IMHO.
That still leaves us with few good options if you have no access to the top of the box.