Privacy Guides

the pejorative "armchair expert" wouldn't apply here as the fieldwork is essentially chair based.

:)

to the same note, you shouldn't upload images of them anywhere. not facebook, not google (drive or any other service), not facebook messenger, but not even anywhere encrypted.

take the images with a regular camera, or a phone that does not give any apps storage access permissions, and have physical prints, which can be viewed every time they visit you. you will need to tell them they can't take photos of the children nor the photo album. this has worked well forag es, they shouldn't be so entitled to images.
if I were you, I would require all guests to leave all their phones on a shelf near the main door. It'll not only prevent photos, but also increase quality time by them not scrolling facebook and such while there.
before you tell them this, let them know firmly that you're not doing this for one, but for child safety and basic human rights, and that in your house it's you who make the rules. and keep in mind, that even when you are the guest, you hold the rights to disallow making pictures of your children until they are old enough to make the decision fur themselves.

why don't use even the private cloud services?
the reason is your relatives who you trusted, will probably download the videos, and reshare them with others through the services you wanted to avoid. also consider that most of them doesn't have any information hygiene, they won't even know they are doing something bad, they won't understand and will hand-wave all your concerns away.
this is not just a technical problem, but also a people problem, which cannot be solved with tech.

if your wife does not cooperate, you won't be able to protect your children to the level you want. of course don't divorce over that or something, it's not worth it, you can probably still do lots. maybe over time, going slower and you can be forming your family's privacy habits.
but I also have to mention, I wouldn't want to live with someone who is not intereinterested in any level about personal privacy. if you have got so far that you're having kids, this is probably not the case for you.

as last words, don't take this as a strong "don't take any pictures" stance. yes, do take pictures, they'll be very good to have later, but make sure that you can keep control over them, for your children's safety.
and don't get (too) mad if parents in the class will take group pictures on which they are there. that's something else, and hopefully relatively rare. best you can do with that is teach your children about why they might not want it, the reasons you don't want it to be uploaded to facebook and such, and that they agree on this they can request the parents to be more careful.