A particular branch of politics doesn't have to wither to absolutely nothing to effectively disappear. Remember it's a winner-takes-all proposal. 49% = 0%. A shift of a few percentage points here and there can effectively destroy a party.
Normally, they have ways of trying to broaden their support base, e.g. in Australia the Liberal Party (neo-liberals) bring in things that will appeal to working-class voters to improve their support, i.e. they target demographics who normally vote for Labor, and do things that will appeal to them, or they give out financial incentives which actually change the demographics into ones that tend to vote Liberal - government cash incentives for new parents is an example: this was literally worded as "helping mothers stay at home with the baby". While obviously nice for new parents to have this money, it was clearly also intended as social engineering to create more stay-at-home moms.
And under Bush, you saw things like trying to get minorities into home loans (The American Dream Downpayment Act and pressuring Fannie Mae to lower underwriting standards). Of course this backfired (big time), but the most likely reasoning is that by turning the Democrat's core constituency into homeowners with mortgages, they could be manipulated to vote Republican in the future based on economic self-interest (the fact that this backfired is not the point).
And I guess that's what the current Republicans are trying to do with the Tea Party and similar, but I think their problem is that they're preaching to the choir: the people they are agitating are people who were already Republican voters anyway, so they're not really gaining much traction. So instead, you see laws put in place to restrict new voter registrations and the like: they want to stay on top by exploiting the fact that their base is older, established voters by merely preventing young people from enrolling. The Republicans will miss out on new voters, but not as much as the Democrats.