I agree. I've never seen есть in a sentence with the format у [genitive] нет [genitive].
Есть is used when the existence of something is significant (or something like that). If something is negated, then the existence question doesn't arise, so есть is not required.
There's an enlightening [and short!] post about existence/есть — and the importance of word-order with есть at:
https: //www.redkalinka.com/Russian-Blog/231/_The-correct-use-of-quotquot-in-quot-quot/ (Despite the silly "quotquot-in-quot-quot" reference in this link, it really does take you to an article titled: The correct use of "есть" in "у меня есть". I tried it in another browser window, and it works fine.)