You've just about answered your question.
Can there be a secure way to allow users to "automagically" change their passwords for different services? Yes...
A standard open API to allow a user to change a web site password is possible and I say it's a decent idea.
Then again, a standard open API to allow a user to log in to a web site is possible and I say it's a good idea. But, amazingly, web site login process is becoming less, not more standardized; this basic function of a password manager is getting harder, not easier.
Well, it makes it easier for a malicious user to cause a whole lot of trouble by changing many of a user's passwords at once. Resetting passwords will be as time consuming as ever. However, it's rare that malicious users change a great many of one user's passwords, even when they have the ability.
The password changer could have its own security flaws too, for example it could choose insufficiently random passwords, and implementations of the API could have security flaws.
On the other hand, it's relatively common for passwords to be compromised in ways that would be ameliorated by more frequent password changes.
So I'd say (with low certainly) that the downside is probably not greater than the upside.
I think a better idea would be to standardize and evangelize a free and open API based on the SRP (Secure Remote Password) protocol, or something like it. With SRP an eavesdropper or man in the middle cannot make many brute force guesses, so strong security can be obtained using relatively short/simple passwords. Furthermore the server does not store password-equivalent data. This means that an attacker who steals the server data cannot masquerade as the client. And if I'm not mistaken, it means that there is less of a problem when users use the same password with multiple web sites.