Unable To Log In To Apple ID on ElCapitan

I am trying to log in to my apple account from a "vintage" MacBook pro. It's running ElCapitan 10.11.6 and every time I try to log in it sends the validation notice to my phone and my watch and I enter the verification code and I get this:

I'm pretty confident it's not a problem with my ID since I can use it from another machine; I mean if the account were locked out I shouldn't be able to log in from any machine right? So it seems as if it's something specific to this laptop. A few thoughts cross my mind:

1.) Is the version of Safari too old? If so is there any way to update it? I ask because I don't have a problem signing into my Apple Id from this browser (Vivaldi 6.7.3329.29 on Ubuntu if that matters).

2.) The battery is not there at all. I don't know if that would make a difference (it's hard for me to imagine how) but I wanted to point it out just in case. As I said, vintage machine and I don't want to spend money on a new battery unless I have to.

3.) What I mainly want to do is to get the most recent supported version of XCode that will run on the OS. Yes, I know it's going to be old but that's fine. Right now what's there is 7.3.1 and I believe 8.2 should work on this OS. I tried finding the area on the apple developers site to download older Xcode binaries but it seems to want me to sign in before it will let me search--at least as far as I can tell.

4.) Just on the off chance that it would make a difference I changed the password. No effect. I also spotted something about waiting 24 hours--I've also done that as well. Didn't seem likely since (as I pointed out above) logging in from a different browser doesn't present an issue.

5.) Oddly enough when I sign into iCloud that works fine--just not from Safari. I mean I can select iCloud from System Preferences and that allows me to sign in (as far as I can tell). But the sign in there doesn't seem to matter to developer.apple.com.

Any help or advice would be greatly appreciated!