Putting this here for my future reference, for when I inevitably hit this issue again. Hopefully others may benefit from it as well.
This morning, I tried a brand-new installation of the latest version of Garmin Express (v7.23.0.0 at the time of writing) on a MacBook Air M2 running macOS 15.0.1 (operating system updated this morning, and the computer restarted as part of the update), so that I could manage my GPSMAP 67 from the laptop. The GPSMAP 67 had never been plugged into this laptop before, and Garmin Express had never been installed on this laptop before. In case anyone tries to suggest that some existing stale data needed to be deleted or some setting reset or the computer needed to be rebooted, refer to the earlier sentences in this paragraph.
Some other threads on this forum suggest that Safari plays a part in Garmin Express's authentication, and that inhibiting cookies in Safari may cause Garmin Express to fail to authenticate. Safari's privacy settings were at their default, and nothing was inhibiting cookies in Safari.
I, too, hit the "Invalid sign in" error when trying to add the GPSMAP 67 to Garmin Express, despite my credentials being correct and tested. I could sign into https://connect.garmin.com/ and https://explore.garmin.com/ and other Garmin sites using the same credentials, copied and pasted to avoid typographical errors, but not into Garmin Express. My credentials were not incorrect, and were not being mis-typed.
In desperation, I pored through the myriad Garmin Express settings, and found a setting, buried deep in the Advanced settings of the application, called "Garmin Connect Authentication Domain", with an empty drop-down menu. When I clicked on it, the drop-down menu expanded to show a list of countries. Having run out of other ideas and with nothing better to do, I scrolled down and selected my own country, then saved the change and closed the Settings menu.
At that point, Garmin Express decided that it would magnanimously deign to sign in successfully, and stopped presenting the "Invalid sign in" message.