I'd wager that "it's my body" argument does not try to hide that part of the debate, but instead declares a clear conviction that the fetus up to certain point is not a person, so it's a statement and it's picking a side in said debate, you can argue about that, in fact that's what most pro-life people do, it's even stated in the name _pro-life_ as in, for the life of the fetus, for that you have to conclude that the fetus is indeed a person.

What differentiates a embryo from a toumor? That it has another person's genetic code?

A mistakenly forgotten medical prop is not a part of anyone's body for example, wouldn't it be fucked up for someone not to be able to remove it from their body if the doctors left it inside just because it's not like their tonsils or her appendix? If there's something I don't want inside my body I should be able to decide, if your argument is that it is not "my body" because it is not explicitly "me" like my appendix, then many medical procedures would also be controversial