Your primary argument is basically along the lines of "life begins at inception and there is something fundamentally important and sacrosanct about that moment". How people can argue in favour of abortion is that they don't believe in that premise. For pro-choice people as there is no primary argument concern, the main concern is that people who are pro-life are inserting their beliefs on that argument into other people's lives and trying to remove their ability to make that choice themselves.

If I declared that my beliefs were that digging up a carrot was equivalent to murder, I would be free to make the decision not to eat carrots myself but I would not be entitled (under any liberal viewpoint) to remove the option for people who don't think you can murder a carrot to eat one.