It doesn't matter that it theoretically all happen in the browser. You can serve different versions to different IPs etc. Every heuristic in me would be screaming don't use that if I would have a need for such tool.
See also https://haveibeenpwned.com/Passwords.
Your question is on target--one I've wondered myself--but I've come to the conclusion that it isn't for people who already have the sense not to put their passwords in random forms on the internet.
I can only assume it has 2 main uses:
1. Poke (some) holes in the bubbles of people with dated password hygiene practices (and a poor sense of how good other humans are at helping attackers reduce the possibility space) by giving them a playground to make new passwords against for a while.
For example, I decided to enter "silverfish3" in the form because I know more than one person who still uses <noun><number> passwords that are multiple characters shorter than this one. It's still turned up in the database 40 times. "dichotomy14" hasn't been pwned yet, but "dichotomy7" has already been pwned 5 times.
You don't have to use a real password of your own to discover that your schema is well explored.
2. I can only hope HIBP password search has scared a few thousand of the kind of person naive enough to fill in the form with a real password straight.