Former USAF & Commercial pilot …
Nomex came into USAF flight suits in the mid-late '60s. Your 1958 model is almost certainly cotton with wool insulation. It may have once had a fire-retardant coating, like kids PJs do now, but that would have washed out after a few washings.
USAF procedure in the 1980s was that all your underwear, etc, had to be non-synthetic. Standard airline safety advice for both crews & passengers is to avoid synthetics as well.
The point to ponder between which clothing to avoid is to decide which threat you’re trying to avoid.
Airplane accidents create truly massive & intense fires. If you are in the fire, whether your clothes are flammable or not is immaterial; you’re a crispy critter either way.
Aviation avoids synthetics because you can easily be far enough from the fire not to get seriously burnt, yet the heat flux will still melt your nylon jacket onto your bare arm. At that same distance, your frayed cotton jacket wouldn’t even begin to char, much less catch fire from the same heat flux.
OTOH, if you are often exposed to small open flame, say bunsen burners, then frayed anything is dangerous. The cuff of a cotton shirt with dangling cotton threads will catch fire about as readily as a poly shirt with poly threads. Heck, I believe that most cotton or poly shirts are put together with poly/cotton thread, so the initial ignition source is equally vulnerable regardless of the shirt’s primary material.
I question that website’s assertion that 100% nylon or polyester is hard to ignite or tends to self-extinguish.
For avoiding flash burns and minimizing the effect of chemical spills, wearing long sleeve shirts & long pants vs short is more important than what they’re made of. Two layers are better than one, e.g. lab coat over long sleeve shirt at work, windbreaker over long sleeve shirt out in public.
If you’re concerned about large spills of noxious stuff, cowboy boots are much better than loafers or sneakers. A sock full of even a moderate acid/base will damage your foot pretty quickly, which can impede your escape. Ditto jet fuel or other petroleum products.
If you have occupational exposure to a specific threat, I’d say wear what’s prescribed as best for that threat. Your non-work life is real unlikely to contain specific threats for which your work clothes are a truly bad choice.
Bottom line:
Absent specific information to the contrary, personally I’d still avoid the plastic over the natural, simply because the natural is easier & quicker to extinguish & a fire doesn’t spread as readily. Natural fibers are much more chemically complex than a synthetic and are therefore IMHO less likely to react en masse to a particular contaminant whatever it may be.
But I don’t know if my opinion is well-founded or just a reaction to a professional lifetime of being told plastic=dangerous, natural=safe, and being safe is job #1.
I just noticed my personal wardrobe is all-but 100% cotton, wool, or silk; all long sleeve or leg; and almost always includes boots. I never bought any of it thinking of safety. I guess that subconscious conditioning stuff works.