A quarter of the drow city of Eryndlyn is openly Vhaerunite. Chaulssin is an entirely Vhaerunite city. Sshamath has an open temple to Vhaeraun. Llurth-Dreir, the biggest drow city, has again a Vhaerunite open presence (and no Lolthites). Ched Nasad, after the events following the Silence of Lolth (which in Arelith however did not happen), became openly ruled by Vhaerunites.
All these can be easily found on both Drizzt Do'Urden's Guide to the Underdark and in the Underdark sourcebook for the 3rd edition, under the respective sections.
Vhaerunites also have a presence on the surface, again open, at least within their own societies, as is the case of the settlements of the forest of Mir. Holldaybim, Iskasshyol and Dallnothax, are all Vhaerunite settlements (with some Ghaunadan presence). They at times war with the Lolthites of nearby Guallidurth, but they do not hide their allegiance. These are mentioned in Demihuman Deities under the entry for Vhaeraun, as well as in Empires from the Shining Seas.
Forgotten Realms Campaign Setting for the 3rd edition also details the Cormanthor drow who happen to be for the overwhelming majority Vhaerunites. Openly so, as they use it both to interact with the elves that live around them.
I am surprised you did not hear of any of these, especially since you also mention Llurth-Dreier later in the post, and that all this information is all also on Forgotten Realms wiki.
I recognized already that they have a strong element in that sense. Their mythologies and ideals are built around the fact that the drow had been corrupted by Lolth at one point, and now they wish to fix the damage. That makes them particularly ideal for antagonists and rebels against the Lolthite drow, as you correctly point out. As you wrongly believe however, that is also a really limiting and incorrect way to assess them. It is quite apparent from the lore and the sourcebooks that if Lolth is not around these faiths can exist without significant issues, as they have pretty precise projects to interact with other people and among themselves. The biggest issue on the server for these is the limit to surface activities and settlements, not that they have not projects or ideas to set in action up there. Eilistraeeans wish to redeem the drow and find a place for them on the surface, but they can redeem them from Ghaunadaur, Vhaeraun or Kiaransalee just as fine, as well as they can fight their worshippers. And at the same time they can try to find a place for their people on the surface, where Lolthites are usually absent. Vhaerunites are pretty eager to do the same with much more violent means, by submitting and conquering whoever they are able to, both above or below, and build their egualitarian, more or less in name, societies. On the surface they also try to reconnect with their forsaken cousins, however usually without much success. For what pertains the activities that these faiths usually do in the Underdark, Andunor allows to perform them relatively fine, i.e. spreading the creed of Eilistraee in secret and helping slaves, or either infiltrating the Lolthites to pursue the cause of Vhaeraun or building an open Vhaerunite faction. Which their players in fact do, and did in the past years, from what I could see.Both Eilistraee and Vhaeraun have a very strong theme revolving around their rebellion agains Lolth. Without Lolth in the setting they would've been pointless, while Lolth would have been just fine without them. They are great for linear storylines (be it a novel or a PnP campaign) where a rebellion can rise up, cast down their oppressors and usher in change. They are also a very poor fit for a PW where such narratives are very rare (as ushering in a change on this scale without a strong endorsement from the Team can be quite challenging).
Here they are at the fringe of what's already fringe, incompatible with everything and a bad fit overall.
Demihuman Deities provides abundant references and descriptions of the designs of the faiths, other than fighting Lolth, so I advise to read also the respective sections there carefully. From the way in which you speak of them, you seem to have the idea that they are the equivalent of Zinzerena, which is just about destruction of Lolthite society with no plans to replace it after it has been burned to the ground. Vhaerunites and Eilistraeeans are much more than that.
One can actually draw the parallel between them and the actual Seldarine, who have Lolth and the drow playing a role in their mythology, and encourage the elves to avoid the practices that turned the drow what they later became. Fighting Lolth is certainly part of what most of the Seldarine stand for, but with the single exception of Shevarash, they have plenty of other things to do if the drow are not around. And so do the other drow faiths other than Lolth, they simply tend to interact with "spiderkissers" more often because that is the majority faith among the drow. The same is true also for Ghaunadan and Kiaransaleean drow, both on the server and in the lore.
Drizzt has absolutely nothing to do with Eilistraee. R.A. Salvatore even commented negatively about the drow deities other than Lolth, so this comes with no surprise. Drizzt seems not concerned at all to redeem the Drow as a people or in finding a way to reconcile being a drow with a vision that does not stem from the Spider Queen. He is a unique individual, again in the view of his author, due to his circumstances of birth, experiences and spontaneous views that nobody really taught him to that extent. His only mild external input for becoming what he did was his father showing a relatively independent spirit toward females, which is absolutely insufficient to explain Drizzt's attitude and behavior.Personally, I have a hard time reconciling with why a drow player character can select a deity like Eilistraee, when players are explicitly forbidden from playing Drizzt clones. While there's certainly more nuance to it, players have been consistently and without fail demonstrating why this is a problem for a very long time now.
It can be safely stated that characters that behave in a way consistent with Drizzt have practically nothing to do with Eilistraee, her beliefs and dogma. Liriel Baenre, the protagonist of some of Elaine Cunningham's novels, is on the other hand the correct example of Eilistraeean drow, until she became Mystran at least. Eilistraeean drow are much more akin to Lolthites and even more so Vhaerunites than to Drizzt. They do not act good because of some spontaneous good nature, but because they simply follow a precise religious belief and they do not seek personal acceptance above, but acceptance for their people. In fact Drizzt is not a follower of Eilistraee, but a ranger of Mielikki.
Lastly I would be rather curious to know how you are able to state with certainty that their players are "without fail demonstrating why this is a problem". Did you interact with every single one of them?