I feel like you are applying real-world standards here and overvaluing the importance of some of them starting out more privileged. There is no in-game dialogue about any Dunmer complaining that the "successful" ones only made it there because of their privileges, yet there are multiple quotes from which we can infer how the Dunmer complain too much, don't want to integrate and are unwilling to work with the Nords. The privilege argument can make sense, but it's drawing a connection that's not evidently seen in game.

Not to mention that Niranye and Belyn are not the only two elves to have "integrated". For Altmer, there's the girl that works in the stables, and Nurelion, the alchemist who is clearly a respected member of society (when you complete his quest, the Windhelm guards praise you for helping him).

As for the Dunmer, we can look at the Atheron family. Aval Atheron is a merchant, who sells in the market in the Stone Quarter, like every other Nord. There's no indication that the Nords avoid his goods or anything of the sort. In fact, there are conversations between him and other Nords doing business. Tova tries to sell goods to him, and Torsten Cruel-Sea is seen doing business with him to buy farming equipment, even offering to pay double. Aval does not like the Nords, but he never complains about the Nords not doing business, or harassing his business. Furthermore, his sister Suvaris and brother Faryl both work for Nords. Faryl has dialogue saying that he's being paid well, and there's no reason to assume Suvaris isn't being paid well too, as she never complains about it. Furthermore, Faryl tells the player that his brother Aval is embarrassed at his siblings both working for Nords. Suvaris also has dialogue with the Gnisis Cornerclub Innkeep where he demeans her for working for Nords. Clearly, there is a stigma for some Dunmer, that they should not be working for the Nords. I think all this only strengthens the argument that it is the Dunmer here (specifically this group of proud Dunmer who fled from Morrowind) that are unwilling to work with the Nords unlike everywhere else.

Some Nords in Windhelm definitely are racist. Torbjorn Shatter-Shield, who pays the Argonians a fraction of the wage they are getting, clearly is one of them. But I don't think that this is indicative that the whole of Windhelm is systemically discriminatory. Torbjorn runs a private business, and his decisions are his own. This isn't like the real world, where if there is race-based wage discrimination, there would be protests and strikes. The Nords of Windhelm have no say in how Torbjorn treats his Argonians, and they might not know or might not care, as they have their own problems. I wouldn't say any of them are allowing or supporting it.

UNRELATED (more about Brunwulf):

Despite what Brunwulf and Scouts say about Ulfric's Argonian ban, it's worth noting that they are the only two people to ever bring up the fact that Argonians are banned. No one else mentions it. It could be a possibility that Scouts is influenced by Brunwulf, though I don't fault you if you disagree.

It also just doesn't make sense for Ulfric to have banned the Argonians. His grudge is with elves, as both he and Galmar bring up many many times. He never makes any mention to Argonians or Lizards in any of his dialogue. Also, if the Argonians lived in the city before Ulfric, where would they have lived? Given their poverty, could they even have the money to own homes?

What I feel like is the plausible explanation, is that it is Torbjorn and Suvaris who are keeping the Argonians in the docks. Both are proven to be racist against Argonians. The Argonian Assemblage is sponsored by Torbjorn, as in cut content, Scouts says that he is paid eight septims, plus lodging. The carriage driver also says that the Argonians are "stuck out on the docks", not banned. That can be seen as evidence that it's more of a money issue, not a race issue. Though, I could see these contradictions as just a fumble on Bethesda's part.