You are on the right track but you don’t see the whole picture, so to say.
The reason why example (1) sounds “wonky” to you is because of “not any” and not because of “there’s not.”
While “not” is used with all verbs, with other words its distribution is somewhat limited:
not all, not every, not many, not much, not often
BUT: *not both, *not each, *not most, *not some (HP 2002: 808)
To answer your specific question, I’ll quote the following passage from the Cambridge grammar of the English language:
“Not any is of doubtful acceptability. Normally one would instead use no or none, as in None of her friends had supported her, but not any is marginally acceptable as an emphatic alternant: ?Not any of her friends had supported her” (Huddleston and Pullum 2002: 808).
This rule applies to anyone, anybody, anything etc. see ngrams
NB: This rule applies to "not any (anyone, anybody, etc.)", with the full, uncontracted "not."