The secret is that restaurants which make traditional cacio e pepe are using pasta water to emulsify the sauce.
But it's not the same pasta water you're using at home!
Only a tiny amount of starch is coming off of the 500g of pasta you just cooked in the proper ratio in 5000g of water (with 50g of salt). They've been cooking with their pasta water all day or all week; It's completely full of starch that came off the other pasta.
Dump a bunch of cornstarch or flour in there to get above 1% concentration (or more efficiently, into a tiny portion in a bowl) to replicate the emulsifying effect, or just use a different emulsifier.
But it's not the same pasta water you're using at home!
Only a tiny amount of starch is coming off of the 500g of pasta you just cooked in the proper ratio in 5000g of water (with 50g of salt). They've been cooking with their pasta water all day or all week; It's completely full of starch that came off the other pasta.
Dump a bunch of cornstarch or flour in there to get above 1% concentration (or more efficiently, into a tiny portion in a bowl) to replicate the emulsifying effect, or just use a different emulsifier.