Disconnecting all the wires from each other was a major blunder.
At first glance, I thought "Oh no. You pulled them all apart. Now we'll never know how they went."
This is actually quite terrible. Because now we're playing "Chinese Whispers" via your recall of how the last guy did it, and that work is not 100% reliable. Normally this is the point where I say "essential data has been lost, time to call a pro with tracing tools and experience".
In the future, simply cut the black and white small leads off the old fixture, and don't mess with the bundles of cable from the walls. Then splice the new leads to the old leads with a blue or gray wire nut.
But if you're sure about what you say about the configuration, that is plausible assuming the "solo" black wire is in the /3 cable. (that would be a modern post-2011 switch loop, actually.) I hope you know which one that is, though experimenting with that particular issue is probably not unsafe.
The deterioration might have been from bulb heat.
The disintegrating insulation on the wires may be from the heat of incandescent light bulbs previously used in the fixture. In particular, many people enjoy ignoring the "60 watt maximum" stickers inside fixtures, and that will cook the wire insulation and even start a fire.
If so, the insulation will be more sound as you get farther away from the fixture - the worst damage will be on the wires going to the fixture. If you find the insulation toward the end of the wire is solid, it'd be possible to save the day with some shrink-tube. You want the better shrink-tube that shrinks 3:1 so it can slip over the old insulation and still wrap the wire decently. A double layer (one over the bare wire and one over the insulation and first layer) would be better still.
Regardless, I would not continue this wire in service without installing an AFCI breaker at the panel, or AFCI receptacle or deadfront at the first opportunity past the panel. Wire arcing is at high risk for starting a fire, and an AFCI gives you the best chance to detect that.
Burning out bulbs
The best workaround to both the "burning out" and the "thermal damage" issues is to fit a bulbless LED fixture. Those generally have a switching power supply which accepts a voltage range from 90V (Japan -10%) to 264V (UK +10%). As such they will not be bothered by weird voltages.
If you believe LEDs still suck like the early ones did, please revisit those assumptions because quality ones are readily available (not necessarily on the bottom shelf at the dollar store, if you get my drift). Regardless please DO NOT put any more incandescent bulbs in whatever fixture goes there, unless it's a chandelier that puts the bulbs far away from the junction box.
However that does nothing about the root problem, which is weird voltages should not be happening, and can burn out a whole lot more than light bulbs. The two most common voltage problems are
- Lost Hot: half the 120V circuits don't work intermittently, and you can force them to work by turning on the electric range or dryer. Hot water takes much longer to recover (but the circuits work while it's recovering). Voltage is low while they're working.
- Lost Neutral: (VERY hard to diagnose): Half the house's circuits are below 120V, and the other half are above 120V, and this teeter-totters back and forth depending which loads are switched on. This is hard to detect without voltage measurement tools such as the ever-useful Kill-A-Watt.
Both of these are supply wire problems with the service coming from the utility. 95% of the time, once you convince the utility that the problem is real, they fix it fast for free. Mine did it in an hour on a Sunday.