Do you use iCloud Keychain? If so, you don't need to copy anything over to your Mac.
I had the same thing going on over the last several weeks, I tackled a half dozen passwords every day or two and whittled them down.. I did all my password changes from my MacBook because I think the 'desktop' websites were easier to do password changes.. My phone has all the updated passwords that were made with my MacBook without doing anything at all.
There is not a faster way to change your passwords. You will need to visit each site and go thru their process
Set up iCloud Keychain - Apple Support