Yes. Still auto-succeed, but you can grant advantage to an ally.
This issue was sort of clarified in Sage Advice:
Can the silvery barbs spell in Strixhaven affect Legendary Resistance?
No. When a creature uses Legendary Resistance, the creature turns a failed saving throw into a success, regardless of the number rolled on the d20. Forcing that creature to reroll the d20 afterward doesn’t change the fact that the save succeeded as a result of Legendary Resistance. No amount of rerolling will undo that success.
Granted, Sage Advice is not rules as written, so let's consider that criteria against the specific questions you asked:
1. Does the spell make the creature roll again, despite having succeeded only due to Legendary Resistance (and if they fail the roll again and have a use of Legendary Resistance left, they may choose to succeed again, thus using two uses)?
No, Silvery Barbs forces the creature to roll another d20 and use the lower:
The triggering creature must reroll the d20 and use the lower roll.
This isn't a new save, it's just adding a requirement to add a d20 and use the lower. Thus if a vampire rolls a 1 they can still choose to succeed.
2. Does the spell happen but effectively do nothing, because Legendary Resistance already declared that "it chooses to succeed instead"?
I'd argue that the spell may not have the full intended effect because it wasn't possible to cause the creature to fail the saving throw due to the use of Legendary Resistance. However, the secondary effect of Silvery Barbs does still occur:
You can then choose a different creature you can see within range (you can choose yourself). The chosen creature has advantage on the next attack roll, ability check, or saving throw it makes within 1 minute.
3. Is it not possible to cast the spell because they technically failed the roll, even if the resolution in the end was a succeeded saving throw?
No, it's possible to cast the spell. The trigger criteria for Silvery Barbs is:
...when a creature you can see within 60 feet of yourself succeeds on an attack roll, an ability check, or a saving throw.
There is no requirement that the creature rolls a success, just that they succeed.
Curiously, this technically would also allow an Evocation Wizard's ally to be a trigger for Silvery Barbs as the criteria for Sculpt Spells is:
The chosen creatures automatically succeed on their saving throws against the spell...
Thereby furthering the argument of Wizards being crazy powerful.
