A more sensible strategy is probably to protect only against large down moves, so get OTM puts, and then longer maturities (like 1/2 year or so), then rotate them every quarter.

ATM options cost approximately 0.4 S sigma T^0.5 (do a Taylor expansion of the "N"s in the Black Scholes formula), so indeed, for current index vols of about 16% we are talking 0.4 * 16% * (1/12)^0.5 = 1.85% for a 1 month option, and 12 of them indeed cost 22% of your portfolio. Not a good idea.

However, if you hold the options only half the way to expiry, you lose only 1/4 of the time value. And if you buy OTM, you have convexity coming your way on the way down.

Lastly, index vols were very low (until yesterday, ha), as so many firms entered the dispersion trade: they wanted to go long dispersion (some firms do well with AI, some lose out), so short correlation, therefore long single stock vol and short index vol. Which means you could buy index vol (ie protection) quite cheap.