This may be medical disinfo. Listerine was long considered a great remedy and preventative care against flus and colds for a long time. This was lobbied against, they were sued and the ftc made them stop advertising as such.
Why would a mouthwash be effective against a virus in your nose?
Using alcoholic mouthwash is bad for the same reason taking antibiotics is - and you may actually want to take oral probiotics for the same reason the ones you eat are healthy. Though I don't know the current state of evidence for them.
Mouthwash is a very American phenomenon though, AFAIK it's not used as commonly anywhere else in the world. Which suggests at least part of its American popularity is due to marketing.
Using alcoholic mouthwash is bad for the same reason taking antibiotics is - and you may actually want to take oral probiotics for the same reason the ones you eat are healthy. Though I don't know the current state of evidence for them.