One word: madras

many words:

wool ties in the summer are definitely a faux pas, ditto wool blended with anything, worn in the summer, for the obvious reaons you state

love a nice wool challis tie in the winter or on a cold fall day, same with a three-season wool/silk Irish poplin, a wool/cotton Donegal in the early spring on a sunny day, wool knit ties on cold winter days can be true friends

but in the summer wool at the neck is devilish, and dry finishes are key, so silk knits are wonderful,  raw slubbed silk is my favorite, cottons, cotton/silk, grenadines, Moygashel linens, and linen/silk ties are really nice, these are the only ties I would wear in the summer, strictly in these hot-weather natural fibers

cotton madras is an amazing cloth, we all agree, and I love it on sport shirts, swim trunks, and walk shorts

nearly all madras ties are plaid, and I tend to think of them more as a weekend kind of tie, they're around, more preppy than Ivy in my view, but there are some that I think look nice, in the more low-key vegetable earth colors, not into the neons or the pastels too much   

I don't have any plaids myself, what I really like are striped raw silk repps, linens, and some cottons, all with a dry hand, also fine with foulards, paisleys, and dots in the hot-weather natural materials, and solid knits as well

regular silk feels a bit wet to me in the summer, greatly prefer the dry feeling of raw silk or linen

Tussah and Shantung silks

trying to remember another bit of old J. Press nomenclature

the labels on my old favorites read "The Burlington Knot for J. Press Inc. Pure Silk Woven in India" and they are slubby repps

and they don't need to always be very candy-hued, for me, just a bit brighter to stand up to the summer sunlight

here is a great piece from 2014 by the one and only Oxford Cloth Button Down on what is my favorite kind of summer tie, the raw silk repp:     

https://oxfordclothbuttondown.com/2014/06/shantung-summer/

here are some more summer ties from Press:

https://www.acontinuouslean.com/2009/06/01/wants-desires-j-press-ties/

Lots of cool options mentioned. Grenadine Fina is my number 1 choice, light and airy. Berg & Berg do nice ones. I've just ordered a Rep Grenadine Fina from HN White, to go with a Hopsack coat I'm having made.
https://www.hnwhite.co.uk/collections/grenadine

Airy knits in cotton, linen and silk or combinations of these.  Slubby silk as well. 

On very hot days when I still want to wear a tie I go to bowties.  No point having the extra layer on my chest.