Zabuza will probably get traumatized due to the Uzumaki reputation. Haku will get traumatized as well. I think everyone is afraid of them at pranks or something.
I can actually see Zabuza not caring about the Uzumaki reputation at all. That's back when it was more common as well. He is the entire demon of the mist. He wouldn't have been worried about them. Haku wouldn't have even heard about them. Long gone clan of ninja? Why mention them.
I can actually see what happens to Zabuza and Haku being a noodle incident whose knowledge of spreads. Zabuza and Haku never ever speak of it. The knowledge of what happened spread at the speed of rumor though. If there were any Uzumaki clauses or such, I could see various old-timers making sure their grandkids had them and were informed of things. It could also be what confirms to other villages that some Uzu are back.
They'd have been expecting the Uzu to bounce back during his mom's generation. The Uzu survivors all married into other villages or disappeared. They aren't gone, they just aren't gathered together as a ninja clan. Now, you've got Naruto and Karin's generation. Most have no idea that they are even related to Uzu, much less knowledge of Uzu jutsu and traditions.
That's where the Uzu Battle Squad is so concerning. It's not one guy happening to survive. It's the survivors rebuilding somewhere unknown and now finally at combat numbers to interact with outsiders. I could see a lot of effort being put out trying to find this hidden Uzu village.
I liked the thing with the various other ninja academies. It showed that ninja/chakra eugenics was an accidental thing. Being born in a ninja village means that you could easily become a ninja. Maybe not without effort, but they'd have the potential and with training start seeing gains. It'd also being in a culture where everyone knows ninja stuff.
Heck, that ramen chief and his daughter might have been able to teach classes just by living in Konoha compared to those other villages. If either wanted to be a ninja or had basic ninja training it's something everyone living in Konoha has. It also looks like anyone who wants their family to become a ninja or get bumped up on the unofficial ninja rankings moves to their ninja villages.
Retired ninja or those that don't make it, still have vastly better ninja genetics compared to untrained civilians elsewhere trying to learn. If Konoha was wiped out, it's not that the land of fire couldn't rebuild the village, it's that it would take generations for the new ninja to be worth anything. Depending on the area, that might have been what actually happened with the remaining Uzu. The civilians tried, but don't have the genetics and couldn't really become higher-tier ninja. The others weren't worried about genin or chunin.
Thinking about it, that might be why some minor ninja villages are minor.
O.k. new theory. When Uzu was hit, their main ninja families and all that died. The surviving civilians couldn't really rebuild it as a ninja village. They don't have the skills or chakra to start with. Also, ninja war was going on so they didn't really want to try then. Now the area where Uzu was still exists as do various civilians. They might not be considered in ninja matters anymore as no more ninja. If anyone from there wants to be a ninja they had to move closer to Konoha or another village. The same goes for hiring ninja. They likely have a nearby ninja office that they need to travel to do their hiring through.
There isn't a need for them to rebuild a ninja village with the ninja system around. Some of the stuff about ninja villages might be national bragging rights as well.