Activation codes are still a thing, but they’re really just for situations like that and are at the discretion of Barclays. I’ve never been given in all my years. It’s always okay we’ll send you a pinsentry. It’s the one banking App I haven’t fully set up on my new iPad because of it yet.
Mobile PINsentry doesn’t work in this case. You need the physical one, because the matching card has to be inserted to generate the correct numbers. I try this every time I upgrade a device. It never works. 
That generate activation code link ain’t working on my old iPad (edit: nor my iPhone. Just throws an invalid link error). Just errors. There’s no option like that anywhere in the app that I can find either. Android only perhaps?
I’m just the messenger here. But I do think it is just for the automated verification process, with anything else requiring a manual review and additional documentation.
I’m not sure why Chase have told you (your wife) any different. They absolutely can’t hold an account to ransom because someone’s changed their name. Definitely lodge that complaint, and if all else fails, the ombudsman. Been a while since my mom upgraded her phone, but I think she just needed her marriage certificate to get back in to Chase. I think Monzo wanted her birth certificate too.