Well some of us don't inherently trust websites enough to direct 60,000 users or their companies clients to them. Entering private authentication and recovery credentials which if monitored/misused may grant access to sensitive data, intellectual property. Then there is potential liability if they are not the full owner of the credentials or it results in further attacks or loss.
No one is talking about inherently trusting websites (such as HIBP), in fact, I explicitly said that you shouldn't just take my word and experience as the complete truth without doing your own research.
But since you still don't seem to understand how this works, allow me to correct some misinformation for everyone's benefit.
By entering your password for checking via the HIBP service, neither your password nor a hashed copy of it is ever submitted outside your computer. Your password is hashed within your browser/application and only the first 5 characters of the SHA-1 hash are sent to the HIBP server. These first 5 characters are then checked against a list of known data breaches and if a partial match is found, HIBP returns a HTTP 200 response along with a list of that remaining suffixes for all password hashes that begin with the same 5 characters as your password hash (as well as how many times that password hash appears in the dataset). You can test this out for yourself and see what this looks like, in fact, I'll use a password that I previously used which was compromised in the Trillian data breach in 2015:
https://api.pwnedpasswords.com/range/B3117Your browser/application then compares the returned suffixes against your password hash and determines whether a match (AKA a compromised password) has been found. For all intents and purposes, this is your offline checking of leaked passwords without having to download, update and compare against an enormous password list.
But again, don't simply take my word for it, take the time to review what other organisations are saying about this useful and reputable service. The API is also very well documented, and since it's open source, you can download and examine the source code yourself from their Github page (if that's your thing). If you decide this isn't something that is useful to you, that's completely OK too.
As someone who works in cybersecurity, it's good practice to check your passwords against existing data breaches using services like HIBP (there are others which I don't use personally so I can't recommend them). In fact, this advice is recommended by organisations such as the US National Institute of Standards and Technology (NIST) under their SP 800-63 (Identify and Access Management) framework.
Myself along with my government and non-government employees have conducted our own assessments of this service and use it regularly for both internal and external client matters.
I hope others have found this advice as useful as I have.