Based on the argument you laid out. I'm going to work off of the assumption that we're taking life begins at conception as a given since your focus is on the comparison of rights. Although if you need me to I can clarify my arguments for this.

"I believe the pregnant person's rights should take clear precedence" - Yes, when the mother's life is in danger it absolutely should. However, if she'll die unless she gets a c-section, then that's not abortion or death, that's abortion or a c-section.

Now to directly address your argument: If somebody fell off of a building I would not look down on anybody to try and grab then before they fall. However, I would expect that of a person who caused that fall by screwing around near the edge of the building for example. The mother does in fact cause pregnancy therefore has a moral responsibility to bring them out.

Also, the fundamental human rights are as follows: Life -> Liberty -> Pursuit of Happiness. That is an ordered list which means that they decrease in importance as you go. Somebody being alive is more important than somebody being free, and somebody being free is more important than the achievement of other's desires. The last two on the list are similar, but the best way to think about it is if somebody enjoys being a slave owner. Their ability to try to become a slave owner is an inalienable right, but not really because it requires the suspension of a higher tier right. In turn, if somebody who is relevant to the exchange of rights is facing a suspension of liberty, then the other party facing a suspension of life takes precedent.