The thing that people don't get about yogurt is that it's a gel, and one with a very delicate structure to boot. The moment you so much as stick a spoon into yogurt you disturb the gel matrix and that causes whey to seep out, an effect called "syneresis". If you've ever tried the kind of yogurt I say above that I'm having for breakfast, you'd know that if you eat say half a tub and then leave the rest for a while and come back to it later you find it has turned into a watery mush that's not very appetising at all. A lot of good yogurt goes to waste like that because folks who want to maintain their waistline will eat some and leave the rest "for later", and it's a balancing act for producers to make yogurt tubs small enough that people will want to eat them in one go and large enough that it makes sense to sell them [1].
One traditional way to deal with syneresis is straining which produces what everyone else calls "Greek yogurt" and we, Greeks, call "strained yogurt" or "bag yogurt" (because traditionally it was strained through a linen bag). However, even strained yogurt suffers syneresis as long as it retains any moisture and as a result you still can't ship a tub of "Greek yogurt" with say fruit, or honey, mixed in just like that [2]. That's why you never get delicious FAGE yogurt with fruit already mixed in [3].
You also can't make yogurt with stuff in it from the get go, because yogurt starts as milk, therefore a liquid, and whatever you want to mix in it will sink to the bottom or float to the top before the milk has a chance to set. Some yogurts with fruit jams for example have the jams on the bottom because that way you just pour the milk on top and let it ferment and then you get jam with yogurt on top and you can scoop both in one spoonful. I've also seen home-made yogurts with fruit mixed in that stay at the top. But you can't just mix the jam or the fruit in the milk and expect it will stay mixed in when the milk sets.
In order to sell yogurt with admixtures without it turning into a goopy gloop you have to stabilise its structure, so that's where the admixtures come in that cause sweetened yogurts to become UPFs. In other words, you need stabilizers. Then you probably also need emulsifiers to "correct" the texture. I don't eat that kind of stuff but I'm gonna bet that the sweeteners also end up being industrial-style ones like high fructose corn syrup or maltodextrin and so on, just for reasons of economies of scale and whatnot. So you end up with something that's many times removed from fresh yogurt that you can make in your own kitchen with milk, a thermometer and a bit of yogurt [4] and that counts as a "processed food" in NOVA terms.
So, no, you can't just ship sweetened yogurt without it turning into a UPF. Or, perhaps you can if you're rally crafty about it, but it's not going to be easy, nor cheap, it's going to turn into a chemical monstrosity anyway, and nobody's gonna bother.
Btw none of the above pertains to Skyr or similar products that I don't know anything about. I'm talking about bovine milk yogurt in the style traditionally made in the Eastern med and the Balkans.
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[1] There's a bit of an exception in ewe's milk yogurt which is much firmer and suffers syneresis more gradually because of the higher protein content of sheep milk. Still, the largest ewe's milk yogurt I used to see was 250g and I don't see them anymore at all though possibly there's other reasons for that. I have never seen ewe's milk yogurt outside of Greece but I believe it's very common throughout the Balkans.
[2] You can strain all the whey out of yogurt. But then you end up with cream cheese, not yogurt.
[3] Btw the scene in Silicon Valley is probably the result of US teaspoons being bigger than European teaspoons. I can totally stick a teaspoon in the little jam tub in Greece.
[4] It takes yogurt to make yogurt. I know. I was endlessly confused about that as a kid.