That's a fun one!
The really succinct answer is "because gravity and electricity can be modeled as linear systems," but that belies all of the fun.
The first thing I'd note is that the force of gravity is a function of all bodies. In your example, the force on $B_1$ is indeed a function of the other bodies: $F(B_1, B_2, B_3)$. The really curious thing about this is that said function of all three bodies can be decomposed into functions of two bodies: $F(B_1, B_2, B_3) = F(B_1, B_2) + F(B_1, B_3)$.
If I may, I'd like to change the notation slightly, to make things clearer. Instead of having one function $F$, whose first argument is the body being acted on, I'd like to bundle those concepts together. Thus $F_1(B_2)$ is the force exerted on body 1 by body 2. The reason I'd like to switch to that notation is because it makes the relationship you are digging at more clear: $F_1(B_2, B_3) = F_1(B_2) + F_1(B_3)$. It gets all of the repeated $B_1$ arguments out of the way.
This is also getting increasingly close to the definition of a linear map. In the definition of a linear map, $f(x+y)=f(x)+f(y)$. This is more or less what we have written above. If I changed the syntax to $F_1(B_2+B_3)= F_1(B_2) + F_1(B_3)$, replacing a 2 argument function taking multiple bodies with a 1 argument function taking the sum of those bodies, then I have the exact same format.
So the short answer stops there. Gravity and electricity can be modeled as linear systems, so we can add these two body interactions one by one, and be done with it.
But judging from past comments and questions, stopping there isn't what you're looking for.
Really all that work above does nothing but move the problem. You asked why forces act like the sum of two body problems. I said it's because they're linear systems. But that begs the follow up question of why are they linear. And that's a good question... with the terribly unsatisfying answer of "we don't know." Science can't answer questions about what reality truly is, only how it can be modeled.
So the first question I would have is, "well, do we see any non-linear systems in physics?" And the answer is without question "yes!" I recently asked a question about the physics of gongs, which in the end distilled down to "gongs sound the way they do because they are terribly non-linear devices."
But wait... all these forces are linear right? Sum them all up? Well, it turns out that all depends on how you choose to model things. A system which is non-linear when viewed one way may be linear when viewed another. It all depends on how you want to approach the problem. In the case of the gong, while all the subatomic forces may be nice and linear, it's not practical to think in those terms when you hit a gong with a beater containing trillions and trillions of atoms. If I think in the practical terms, where I have "one beater" instead of trillions and trillions of atoms, the gong behaves nonlinearly, and I live with that.
Science, as a whole has found that the deepest underlying behaviors of our universe tend to be linear, when viewed in the "right" way. This proves terribly convenient. In your example above, imagine splitting $B_2$ into two halves, $B_{2R}$ and $B_{2L}$. It's massively convenient that $F_1(B_2) = F_1(B_{2L} + B_{2R}) = F_1(B_{2L}) + F_1(B_{2R})$. This says that nature doesn't care how we draw our lines, dividing objects up into pieces. The resulting physics is the same whether we have one object, or two half objects, or an infinite collection of point objects integrated together. It's a strong nod to the dual of the anthropomorphic principle -- a nod to the idea that nature is not defined by us... nature is nature.
Or is it? The deep question you scratch at when you question whether gravity is linear or not is whether the underlying nature of our universe is as linear as science thinks it is, or if it actually isn't. And that crosses the line into philosophy, but I think it's a good line to cross (even if all you do is tell them you're just visiting and have your passport stamped).
If you really think about it, that's the kind of questioning they have at the LHC. Okay, fine. They're a wee bit more formal in their questioning, and their knowledge of subatomic physics is far better than mine. But its in the same class: "We see behaviors X Y and Z. Are these behaviors the fundamental nature of reality? Or is there something deeper that we haven't seen yet?" After all, one of the things you'll find in the study of calculus is that nearly everything can be linearized within a small range. But step outside of that range, and the rules appear to shift. The LHC literally dumps gigajoules of energy into trying to find where those rules might shift, and to capture what happens there for scientists to plumb for the next deeper question.
But even they have to deal with the fundamental reality of science: you can't measure what you can't think of. If you're not looking to measure something properly, you'll never see it. And we do believe our human brains are built of the same stuff the universe is made of, so it has the same limitations. My personal favorite capture of that appears 13 minutes into the VSauce video How to Count Past Infinity, where they do an excellent job of capturing the difference between the mathematician and the scientist.
And, of course, there's still the philosopher. One step beyond physics is our study of metaphysics, and that is always outside the realm of science. There is nothing in science that can disprove the claim that we have a soul, and it is special, beyond the realm of physics. 'Course they can't prove it exists either. But that's the fun of that realm, and there's a whole 'nother class of inquiry that follows once one considers something outside of the laws of nature.
