I've had my Visa card for a very long time (decades). I've been happy with the provider, and the few times I needed the weight of Visa behind a dispute, they came through. No fuss, just like I want a credit card to be.

A few months ago they started sending me email to invite me to add another authorized user to my card, suggesting it as a safety net (so if something happens to me, someone else can administer my account). Maybe that appeals to someone, but I'm not interested so I ignored it. More recently they have been offering minor inducements (a one-time small credit) to do this, and that makes me wonder what their real goal is.

If this is merely a service they offer for peace of mind, the peace of mind is the inducement and nothing else is needed. That they are trying to entice people to do it means there's some other motivation that benefits them more directly. I'm assuming this is not a way to add your minor children so they can more easily make in-app purchases or whatever the kids are doing these days -- and anyway, unless they're giving you a way to throttle spending from other users, that would be a very bad idea.

The only thing I can come up with is that this is a way for people with bad credit scores to get access to credit cards. They aren't going to issue cards to such folks directly, but if they can get you to add your deadbeat cousin with a terrible credit rating (to "help" your family member), then the credit-card company gets more transactions and thus more transaction fees at very low risk to them. They know an existing customer who'd like to keep a good credit rating is on the hook for the charges; they're going to get paid. This might be in Visa's interest, but how is it in mine? It's not, which is presumably why they're trying to buy folks off.

Have I missed some benign reason for them to push this scheme?

(Still not doing it, but curious.)