By claiming that a gay judge can't make a ruling on a gay issue is essentially saying that gay judges aren't capable of being impartial. And if you're saying a judge isn't capable of being impartial then you're also essentially saying they aren't capable of being a judge.
If the R-71 folks truly did feel that a gay judge was capable of making a ruling then they'd have no problems with this judge as there'd be no reason to think she couldn't be impartial on the issue other then the fact that she's gay.
No - by claiming that an ACTIVIST gay judge can't make a ruling on a gay issue is essentially saying that ACTIVIST gay judges aren't capable of being impartial on CASES INVOLVING GAY ISSUES, not all cases generally. And if you're saying an ACTIVIST judge isn't capable of being impartial on ISSUES ON WHICH THEY ARE ACTIVISTS, then you're essentially saying the judge is BIASED WHEN JUDGING A PARTICULAR CASE, not incapable of being a judge generally.
IF the proponents have actual evidence of the judge's activism (and, as I said twice before, it doesn't sound like they do), what they are asking is no different than if a school funding referendum were on the ballot and the judge vociferously protested or signed a petition against it. Should he or she recuse himself? Of course. Does that mean he or she is unfit to judge in all cases? Of course not.
The fact is that parties ask judges to recuse themselves over actual or perceived conflicts of interest all the time. Sometimes they do, sometimes they don't. That their impartiality in a particular case is questioned doesn't make them unfit to judge generally.