This should not be an issue.

When you complete a user-login (credentials + 2FA or QR-code), Steam sends a login key which acts as authorization token to show Steam that you know how to log in. This is used on further logins, so you can log in without entering anything at all.

In order to do that, the client stores that login key. Thus, if it doesn't have a login key, you might have disabled it entirely -- settings have an item for that in the "security" tab. If you've messed around with filesystem permissions, it's also possible that it simply can't save it anymore.

Thanks for the reply but I still don't understand, really. Or I suppose I understand that there are many factors that could cause this problem. I've heard of 2FA but I've never noticed that I need it to log onto Steam.

Ah yes you're probably referring to the security code (or "code") I mentioned. But at the time of writing things do not seem to be working.. and the word "layers" come to mind for me. I guess I just DON'T know how to log in. .

For some reason I suppose at least the iMac can't store the login key you mentioned. I might, be wrong and I'll look into the security settings you mentioned.

Edit: OK it seems that Steam did not have "full disk access" in the iMac settings... I changed that and at least this time I got a new type of error message after I entered the verification code: "there was a problem logging you on to steam -- try again?"

Okay, I should be more accurate. The actual error message I get is: "Something went wrong while attempting to sign you in. Please try again later."

...Wonder what that might be. At least I seem to be getting closer to working this out. :/ It's better than Steam auto-quitting on me while I'm typing in the required info. Yes that actually happened.

Sorry for the double post.