When do foetuses become "people"?
That depends how one defines people. Everybody who was ever born is or was a person to me, and I never refer to the unborn as people, although it's not important to me if others do. It changes nothing with me, including the moral status of abortion. As I use the word person, abortion isn't an issue. People have already been born. If others care to use the word differently and apply it to the unborn, it won't affect my view on the moral status of abortion. No words do.
And I'll bet that most people who think about it will agree. Consider any act that one considers moral or immoral, and see if changing any words without changing what they describe matter to your moral assessment. Call a eight-year old girl an adult woman, and see if that changes how you feel about her being married or being sexually active. Your question is analogous to asking when she becomes a woman, and the answer is the same - that depends on how one defines woman, but however you do, intercourse with her is immoral if she's prepubescent.
so why isn’t the fetus/embryo a person? (regardless if it's ok to kill it or not)
I still don't know why persist in this. Go ahead and call it a person if you like. Call it the king of the universe. Issue it a Social Security card and and a passport. Give it the right to free speech and to bear arms. It doesn't matter. If it's presentient, it can be ethically aborted whatever you call it.
The abortion debate is not an issue of killing child vs killing an embryo
I disagree. It's about how they are different, and why only one is moral because of that difference. And it is about who gets to decide the fate of the pregnancy, the potential mother, or the church using the power of the state to enforce its theocratic proclivities against otherwise autonomous citizens, which makes it a First Amendment and Fourth Amendment issue as well.
I am just presenting adoption as a better solution than abortion// so ether agree or refute this point
It's an opinion. It is neither correct nor incorrect, and thus can't be proven or disproven (refuted). I prefer abortion over adding another unwanted baby to the roles of foster children in an overpopulated world if a woman doesn't want her baby. You can't refute that, either. You can only say that you feel differently. You value one more life on the planet whatever its fate, others one fewer.
The claim is that the fetus is worth more than the woman’s desire to avoid 9 months of discomfort
And that is your opinion. It informs your choices. But her pregnancy has nothing to do with you. It is her opinions that matter, and she needn't justify them to you or anybody else. Personally, if I were a woman and found myself pregnant, I would want an abortion for a variety of reasons, none having to do with discomfort or inconvenience.
That's the anti-choicer's framing of a deeply personal decision with significant ramifications if allowed to go to term that would cause someone like me considerable regret. What if I can't give it up out of a sense of responsibility and I have to raise a child I didn't want and possible can't afford? If I give it up, what will this child's life be like? Will it be tossed about the foster care system until ready to take its revenge on society? Will it be gunned down in a church or school? Will it become a climate refugee? Will it live under tyranny?
This is why YOU (and the state on behalf of the church and its dogma) should not be the ones to make these decisions for others, and why humanists advocate for choice. You and they have no interest in her, or in the fate of the child - just that your deity be served with yet another live birth however unwanted by its mother. You have no argument with those who don't share your religious or other beliefs. You can only say what YOU would do, not what others should do, and hope that you can persuade them to agree with you.