If I supported them in general, I would also be lending my support to that policy.

Yeah, no, hard disagree there. My opinions and values are not shaped by the person I vote for, it's the other way around.

Reality is more nuanced that what you're describing.

For some people, the denial of basic human rights is a line in the sand, and any support for that kind of policy (no matter how tacit) is a major red flag.

It's a line in the sand for everyone. In the instance of Roe v Wade (which is what I assume you're referring to) both sides of the argument believe they're fighting for human rights... one side just believes the rights of the mother are more important and the other side believes the rights of the unborn child are more important. It's a matter of perspective, and from the perspective of people on both sides of the aisle the other side is morally bankrupt and they themselves have the moral high ground.

The issue is that the problem simply isn't as black and white as people want it to be. If there was a clear and incontrovertible answer to who was right in these debates there would be no debate.