As a counter-example, consider the "9" chord, has an added 7th, and an "add2" chord, which doesn't (despite editors continually writing "9" chords where they should be putting "add2").
Similarly, a "sus4" chord doesn't have a 7th, but a 11 chord does.
As for "jazz"
vs. "blues", I thought the rule was that blues only uses chords up to a 7th. Anything higher than that - in
any musical genre - and it's automatically "jazz."

Speaking of
Spooky, don't forget the intro to Carole King's
Only Love Is Real. Great song, but the intro is...
Spooky.