Toothpaste is not a scam.
Mouthwash really doesn't do anything by most accounts, and in some cases does harm by affecting helpful mouth bacteria while supposedly trying to kill "bad" mouth bacteria. Mouthwash should be thought of more like makeup, it might make you feel good, and so if it does, go for it. Is confidence or a "Fresh feeling" health-related? In the same way a haircut or lipstick is -- sure, why not.
The jury of medical science is out on whether "Listerine prevents gum diseases." It's an alcohol-based solution, so it'll kill some organisms just like using an alcohol cleanser on on your acne might, but the idea that it's actively fighting against gum disease and doing so effectively is a marketing exaggeration. Don't mistake marketing slogans for actual health recommendations.
Listerine has a wonderful history of claiming to treat things as a miracle cure that it's never been proven to do anything to. It was originally used to clean floors, which sure, it did a fairly okay job at like any alcohol-based cleanser; Then it was marketed as treating gonorrhea, which it didn't work at all, then the common cold, and everything else, until finally Listerine basically made up the concept of "Halitosis," the word was basically unknown in english, and they hit on something a lot of people worry about but probably don't need to -- chronic bad breath (which listerine really can't treat at all, as most chronic bad breath is way past your mouth)
Incidentally SYSK reposted their ep on Halitosis a couple weeks back:
https://www.stuffyoushouldknow.com/podcasts/sysk-selects-halitosis-worst-smell-ever.htm