You stated the wiring existing was BRITTLE.
This IS the problem, plus those old fixtures originally had a warning that said 60 watt max type A bulbs so using 100 watt bulbs only increased the trouble!
The whole thing is like this:
The 60 degree C and 75 degree C and 90 degree C ratings refer to the temperature (ambient) that the insulation can protect the conductor safely. That old TW and old sheathed electrical cable wire was 60 degrees C (or possibly rated at 75 degrees C for over 100 amp applications).
The HEAT generated by an incandescent bulb is HUGE compared to the light "candles" it produces. Its engergy requirements similarly imbalanced, hence making it a very inefficient lighting source.
The problem is JUST WHAT YOU DESCRIBED THAT YOU FOUND!! Brittle, dried out (that old insulation was rubber based), sometimes burnt, sometimes oxidized insulation and wiring. THIS IS A SOURCE OF an ARC-FAULT and or heat sourced FIRE. YES you are at risk, YES you have a problem, and YES the rest of your home where other such wiring situations exist are at danger. This danger increases over time.
The USEFUL life of your cable was 20-25 years as originally designed, assuming it was NEVER EXPOSED to over-voltages/amps and excessive heat, and other such age-reducing hazards.
The HEAT generated by 3 x 60 watt bulbs in the ceiling fixture, let alone 100 watt bulb(s) PLUS the ambient temperature of a box and cabling in the ATTIC (without added burden of heat generated by the lightbulbs themselves that's already over 51 degrees C) your cabling probably has aged expidentially. Yes you need to inspect your wiring, it is the insulation that is at greatest risk and if brittle, it isn't protecting against arcs between the hot and neutral in your sheathed electrical cable itself!
In general, if the cabling is a-okay, yes it is possible to "pig-tail" to a box the existing 60 degree (ASSUMING ITS GOOD/SAFE) then junction to 90 degree and run the 90 degree to the FIXTURE box, but each and EVERY bit of wiring to that fixture box, coming and going and THROUGH that box must be 90-degree rated to its next point at 3-plus feet away.
Frankly I would't trust 30 year old sheathed electrical cable in an attic for any reason, its already beyond its "useful life", as originally intended. I'd replace the entirety with EMT enclosed THHN of the correct rating for distance, derated for conduit and attic.
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