There are two '66, 289 Mustangs in our garage. One has all its original harnesses but the other has a few AMP repros to replace wires burned out from an engine fire long before I bought that car. We drive the original, A/C coupe on long summer trips every now and again to the midwest and beyond (VA) since that is where we lived until age 21. The coupe is the one with the halogen Cibies since it is more likely to be driven at night, far away from home. My first new car was a Fiat 124 Spyder which now has a similar set of Cibies.
I admit, I piled on the words.
That thread provides a Purdue EE's analysis of the issue which, I admit, goes against common belief. Common does not mean correct. My version uses the laws of physics to disprove mass opinion. Halogen, by itself, was not your problem at all. Halogens are used for more and better light using the same, or less, electricity than the prior incandescent bulbs used. LEDs are more efficient yet but to generate a lot of light they have to use an AREA emitter and not what is essentially a tiny, point or line source used for halogens. The optics to use LEDs are vastly more complicated and different and you won't find any decent ones for $30 or in any retrofit kit. You may find some LED bulbs that splatter out a lot of light but I have not seen one that produces a decent light pattern for a retrofit application. And keep in mind that high power LEDs run incredibly HOT which is why they have huge heat sinks on them. Especially the el cheapo versions may not be designed for a very long lifetime.
Now, if there were 100 W, or larger, halogens in your car the story would be somewhat different although not greatly. A watt is a watt no matter how its generated. 100 W halogens will always be marked Off Road Use Only since the legal versions will be 55-60 W.
Some of the wiring harnesses you might find today can have really small, inadequate gauge wires; I have seen that problem myself although not with Mustang repro harnesses made by AMP. Undersized wires will cause problems now and forever. I have even seen what were called jumper cables made with 12 ga wire. Useless, except to charge a cell phone since a car battery cable is usually a 2 ga wire or larger. Every starter uses a relay, i.e. solenoid, since starters require a huge number of amps that small wires cannot supply.
Your profile says you are in Alabama? I never expected you to be near Looavul. I grew up listening and watching WHAS and even had my 5 minutes of high school fame in an interview on WKLO for some now-forgotten reason. In those days it was an AM station in downtown Louisville and was NOT into country music.