Branding and conservatism
One problem that I think would crop up with your idea for a non-branded group like the "League of Women" is that it would swiftly and invariably become known as "the conservative League of Women" in media accounts, and it would be lucky to avoid "ultra-" before "conservative."
The media largely controls this branding business, no matter how individual groups choose to present themselves. We all know how it works - it's tough to find any group described as "ultra-liberal," isn't it? Lefty groups are described as public-spirited or passionate about their causes; right-leaning groups get tagged with more bluntly political terms.
Coverage of these groups is also divided according to political bias. There was a story going around earlier this week about how the far-left Tides Foundation not only escapes being described as "far left" or "ultra-liberal," but barely gets mentioned at all; it's allowed to conduct its business in secret by the press. Tides throws around five times as much money as the (in)famous Koch Brothers, but the Kochs get 15 times as much media scrutiny. If your label-free but right-leaning League of Women took so much as a dollar in Koch money, even indirectly, it would become known not only as "ultra-conservative" but also "Koch-funded," just about every time its name was mentioned.
This is one of the greatest powers still retained by the mainstream liberal media. Conservative media has achieved some inroads by successfully exposing or labeling a few groups here and there, but it's very small potatoes by comparison. Conservative groups are probably best off anticipating it and smoothly responding to the labeling by turning conversation back to their key mission points.
As for the public sense of anti-conservatism or anti-liberalism, it's clearly different among different groups, but among that great formless mass of swing voters, I suspect they'd be more likely to insist they're "independent." The funny thing is that they reflexively equate "liberalism" with independence, when the modern liberal creed is anything but. They may also confuse social liberalism or libertine hedonism with independence, when in truth economic liberty is far more important. The crumbs of personal license celebrated by the Left are nothing compared to the real freedom of action they take through mandates, regulation, market interference, and seizure of wealth. (Time is money, and all that.) Maybe if conservatives emphasize more strongly that liberty is what they're trying to conserve, the label will regain a share of whatever appeal it has lost.