The thing about "lol supertech from vibranium" is that there was also a a line somewhere in the comics about Wakanda being a nation of geniuses. Also, it is a brainbug. If you go back to the first depiction of Black Panther, this wasn't the case and it was the Black Panther all by himself that modernized the country relatively recently. "Wakanda is a supertech utopia made by black people" is an idea that developed and matured over time.
But Wakanda rather obviously is an African-American's utopic fantasy of Black Unconquered Nation that superseded their own country in most respects (except democaracy, which was actually a topic of a civil war in current comics). It is a narrative that is counter-racist (outside Wakanda, most comics would have unlikely to have features a city of superior technology that wasn't made by whites or aliens) that sadly still works on racist logic (race is key, Wakanda MUST mean black). In the comics there was an emphasis on actual science and engineering, with T'Challa also being a scientist that was Tony Stark's equal as well as a strategic genius. The idea that Wakanda is a country where the "black" race was never "held back" by outside influence, you'll see the superior race undertone easily lead to "nation of geniuses".
That said, I do have one explanation that is partially supported by the canon: Bast, plus the rest of their gods (they have their own pantheon, can't be bothered to look it up atm) influence. They take a present role with the country. Wakanda wasn't just depicted as technologically superior, but spiritually superior. The heart-shaped herb wasn't the Wakandan equivalent of a super-serum. It was used to help one reach commune with Bast herself and it is her that grants the power of the Black Panther (which included supernatural powers). If you did not meet Bast's approval, you died. The Black Panther wasn't just king, but the pope of Wakandan religion as they (there were female Black Panthers) represented Bast (all the statues? They were for Bast, not just the king).
The herb was supposed to be the final test for becoming Black Panther (the challenge thing was actually meant to be public event where anyone could challenge the king, thus ensuring that the king was always physically capable). This goes further, with the nation retaining supernatural knowledge as well as scientific, or with little separation between the two. There are actual priests and shamans of supernatural power in Wakanda on-call.
From this, and this is where my speculation starts, you can see a similar attitude made for education as well as clear respect for knowledge. Knowledge has its own god, Toth IIRC. Consider that Wakanda's intelligence network didn't just take information about potential enemy's intentions, they took information about technology as well**. Combine that with a strong, spiritually-enlightened focused on education, the fact that vibranium is universally useful and that Wakanda is very rich (not just by Vibranium although I don'T recall how), and you have the basis of a virtuous cycle of technological development.
Of course, there was little time for all this in the movie. I would not be surprised if later on we see deleted extended scenes about Bast or that the religious aspect was taken out almost entirely except as background detail.
If Captain America's shield is literally the ONLY openly known piece of vibranium in the world, beyond perhaps the occasional artifact or two (depending on how you read museum lady's reaction to Killmonger declaring that the weapon in the case was vibranium), that raises the question of why scientists haven't tried to test the properties of vibranium further. One would think that an artifact or two might have been sacrificed for the sake of science. It also raises the question of why exactly Howard Stark used a random unknown metal to make the shield. He probably had at least some idea of its capabilities. I don't know how far 1930s-40s Western science could have gone with that.
In the comics, the current Black Panther's grandfather gave it to Steve Rogers after he helped repel a Nazi invasion to Wakanda (Wakanda gets lots of invasions). Depending on the comic, the Black Panther actually BEAT Cap (Cap was still inexperienced at that point).
How it got into Stark's hands in the MCU is unknown. All I can think of is that at that point the Allies have gathered up lots of weird stuff that the Nazis gathered up and Stark got his hands on all that. Then, being Stark, he experimented with it and produced the shield among other things. I do recall that Stark had several shields ready for experimental purpouses.
** Someone pointed out how a black person having access to delicate and secrative scholars would be strange in various times. Well, there are two simple answers to that: first, Wakandan spies had superior technology that likely included things listening devices; second, Wakandan spies could have simply made or gotten white (or Asian or whatever was necessary) agents that would have done the interrogation for them.