What is the physical reason for why gravitational potential (or electrical potential) due to two masses at a point can simply be added algebraically?

It looks like you are confusing vector magnitudes with potentials.

The potentials are indeed additive because the forces are additive, which is confirmed experientially.

If $\vec F=\vec F_1+\vec F_2$, and we know that for a conservative force $\vec F=-\frac{dU}{dx} \hat x$ then we have

$$-\frac{dU}{dx} \hat x=-\frac{dU_1}{dx} \hat x-\frac{dU_2}{dx} \hat x$$

Then we can integrate both sides with respect to $x$ to get

$$U=U_1+U_2+U_0$$

Where $U_0$ is some constant. So you see the potentials do add given that the forces add, which it seems like you agree is true. This argument works in more than one dimension as well. Notice also that none of this depends on how the magnitudes of $F_1$ and $F_2$ compare to each other or their sum.

We can also use what you reference about doing work from infinity:

$$U=-\int_{\infty}^O \vec F \cdot d \vec x=-\int_{\infty}^O \vec F_1 \cdot d \vec x-\int_{\infty}^O \vec F_2 \cdot d \vec x=U_1+U_2$$

If you think about it these two methods aren't that different. The latter one just determines what $U_0$ is by setting the potential to $0$ at infinity.

Point O is a point of unstable equilibrium. This means that the potential energy is actually at a local maximum at point O. Forces just determine the slope/gradient of the potential energy, not values of the potential energy. This is evident from the above work, and knowing that you can always add a constant to the potential energy without changing the physics.