Your last sentence is the biggest problem for most anti-choice people. They also generally believe in the conservative bootstraps ideal. So not only do they want to prevent a poor potential mother from having an abortion, they also want to make her solely responsible for raising a child, and then also ultimately punish the mother or child when/if they make the bad decisions that some poor people make. In many ways, the USA is a political clusterfuck because the supposedly laissez faire right wing takes a hardline on issues like abortion but provides no backstop at all for them.
Realistically, while I am against banning abortion, any ban should also come with a fully funded nanny state that allows the mother to fulfill her dreams while also paying for the child to be raised. This needs to go far beyond SNAP and TANF, and needs to include day care and tutoring at the very least. We also need to reform the adoption system that essentially "sells" babies. Adoptive parents shouldn't have to go into debt to adopt a baby. That money should go towards raising the baby.
However, this idea that the potential child is a part of the "people" and not just the child of the mother is something that the USSR believed and it was why they banned abortion up until 1955. But even there, it failed, as the USSR had a very high illegal abortion rate prior to 1955 and the highest legal abortion rate in the world after 1955. At the end of the day, for a potential mother, whether a baby feels like an anchor to her isn't only, or even mostly, based on finances.
As for your first 3 paragraphs, fetus is an old word. All of what you say is just semantics. There are multiple bright line events that occur in the growth of a fetus. Heartbeat. Brain function. Viability outside the womb. We could probably find an agreeable cutoff point. Yet everyone fights this fight in both directions with the "if I give an inch, they'll take a mile" strategy.
But also, it should be a women's choice. And there's a reason that women largely support abortion remaining legal. Its entirely on them. Humans are a fragile species. Our young aren't truly capable of surviving on their own for roughly the first 10 years of life. And the fact that humans not only still exist, but thrive, is due entirely to women taking the responsibility of motherhood seriously. Yet ever since we were living in caves, some women probably weren't ready and threw newborns off the cliff. So while men like you and I can certainly have opinions, and we can save newborns from being tossed off the cliff these days, the decision on whether or not to allow another lifeform to grow and feed inside someone should fall on the person who has that lifeform growing and feeding inside of them.
If/when we invent an artificial womb, then perhaps others can or should be involved in the decision making process. Yet even if that occurs, I imagine we will then be looking at a significant number of people that follow the artificial womb/orphanage/prison route because the only form of socialism that US conservatives seem willing to fund is the prison system.