Is Vespasianus really remembered for the Vespa?

No, he was the Roman emperor who introduced charge for public toilets.
Rome was in bad shape when he started his reign. Nero had spend a lot of
public money, had fun watching Rome burn, etc. So ir was time to get the revenues up.
Good bye to tax exempts, hello to pay toilet.
The French still call public toilets vespasienne.

By the way I cannot say that I purchased the Vespa exactly 1900 years after Vespasianus.
This is because July 1. in 69 is not the same day as July 1, 1969. Those days Julian calender was used. That is for Julius Caesar , played, for example, by Marlon Brando.
The year was actually 11 minutes too short, so when the calender was changed to after 1600 years of use you had to jump the date by some 11 days or so. People thought
that te government stole these days fom their lives.
Another thing about old Julian calender. The two Russian revolutions are known both
as February or March  revolution and the October or November revolution. This is
because they were pretty old fashioned and still using old Julius' calender.