You forget the rights of who is in that reproductive equipment, and whether or not who is in there actually counts as a person.
The only one who "counts as a person" is the woman. Not the fetus. Unless and until there's a constitutional amendment providing the rights of citizens to pre-viable fetuses.
As I've said many times before, this does not prevent folks of good will from persuading and supporting women who do the right thing. But this is NOT - from the legal perspective - a case of one person's rights conflicting with another. A pre-viable fetus is not legally a person. A pre-viable fetus does not have legal rights vis a vis the mother. It is a part of the woman's body over which she - and she alone - has dominion. The Constitution rightfully protects such dominion from coercion by the State.
And, yes, the protection of one's individual liberty and autonomy from coercion by the State is fundamentally conservative. And to demand that State exercise coercion over a woman's fundamental decision whether to reproduce is authoritarian, not conservative.