It needs to actually be clarified that fertility circles misuse the term embryo. Truth be told, I actually do believe real embryos are the tiniest beginning of people. But only when embryo is used correctly to refer to post implantation. Pre implantation, we have a fertilized egg, a morula, finally a blastocyst, depending on number of times cell division has taken place. Mere cell division doesn't turn a fertilized egg into a baby. For that, you need the womb. Only when planted in the lush uterine wall of a woman does a seed stand the chance to grow into a seedling.
Furthermore, the fact that before implantation, a blastocyst can split into two separate blastocysts, possibly resulting in the implantation of both and resulting in identical twins, this should tell us personhood doesn't begin until we are certain only one individual will result. Not only that, but two blasts can also fuse into one, forming a person known as a chamera, or an individual with two sets of DNA. This should prove that a singular unique DNA doesn't equal an individual, as is commonly thought.
This has little implication for the pro-life movement when it comes to abortion, since many women don't seek abortion until they miss their period, which means implantation has already taken place and - according to the above reasoning - there is a tiny baby at stake.
But this reasoning does have implications for contraception-as-an-abortificant that is cited as one reason against contraception by the Catholic church. Birth control pills can indeed prevent a fertilized egg from implanting, but if we were to accept implantation as the point of personhood, this should have no bearing.
This reasoning also has implications for in-vitro fertilization. Many people who complete their fertility treatment and are left with pre-embryos they do not plan on transferring in hopes of another baby are faced with a dilemma, at least those who believe that life begins at conception. They don't want to keep the pre-embryos for themselves, either by transferring them or paying storage indefinitely. But they don't want to destroy them or donate them to science either, since this would be synonymous with murder for those who believe a fertilized egg is a human person. The third alternative is to allow the pre-embryos to be "adopted" into another family who will transfer them and hope to grow and raise their own baby. Sounds like the perfect solution, right?
Yes and no. See, while pre-embryos are in the pre-implantation stage, it's one thing. But once they are implanted and develop into a true embryo and later fetus, there is no going back - there won't be two out of one or one out of two. However many implantations took place, that's how many babies are growing. And if those babies have DNA from a family other than the one they are being born into and raised by, then issues of identity and medical history and general access to the child's genetic relatives and ethnic heritage become issues. These are not things that can easily be figured out after the fact, and yet many people don't think about the potential future child's sense of self when choosing fertility treatment options. I'm not saying third party reproduction should or shouldn't happen. I'm saying that when it does, the potential child's best interest needs to be paramount, both from the perspective of the donors (that they be willing to be known, at least when child turns 18), and from the parents (that they be open with child about their background and willing to support contact with genetic family).
I don't expect us to all come to an agreement on when human personhood begins any time soon. People make rationalizations for killing each other at all stages of growth and development, way past the in-the-womb stage, so there will always be those who simply don't care about the experiences of the in-utero fetus.
A side note, I used to cringe at the term "fetus", as I thought it belittled the humanity of the unborn baby. But I realize now that it's merely a stage of development, just like newborn, infant, toddler.... Interestingly, when I was at a store with my newborn daughter, a teenage girl approached us and with a squealing voice announced, "what a little fetus!" I was horrified, but in retrospect I see that she clearly wasn't saying that my daughter wasn't a human being! She was saying that she was so small she could easily still fit in the womb, which was true, as she was born weighing less than 6 pounds!