Normally, we see spherical coordinates defined as follows: let your point $P$ lie on a circle of radius $R$ with centre at the origin $O$. Let the angles made by the projection of $OP$ onto the $xy$-plane and $OP$, with the $x$ and $z$ axis be $\phi$ and $\theta$ respectively. Then the coordinates of the point are, $$(x,y,z)\equiv (R\sin\theta\cos\phi, R\sin\theta\sin\phi, R\cos\theta).$$
I see that sometimes we need to consider pieces of the sphere, "differential area". It is calculated by the equation, $$\text{d}A=(R\text{ d}\theta)\cdot (R\sin\theta\text{ d}\phi)=R^2\sin\theta\text{ d}\phi.$$ The idea is that a rectangle with side lengths $\ell$ and $b$ has area $\ell \cdot b$ and we can approximate the differential area as a perfect rectangle with second order deviations (that do not matter as we take smaller and smaller pieces). However, I have been skeptic with this reasoning, because it can be deceptive. Some examples where there are holes in the geometric reasoning and that lead to incorrect conclusions are discussed in a 3b1b video titled "How to lie using visual proofs".
I want to know why this deduction can safely be made. In particular the main issue that comes to my mind is that the formula $\ell\cdot b$ assumes that the side lengths are perpendicular to each other. If we see the differential area from "top view" (don't neglect the curvature), the correct direction to go seems to be perpendicular to the motion of $\hat{\theta}$ (specifically to the line segment formed due to $\theta\to\theta+\text{d}\theta$) and not along $\hat{\phi}$. Initially I thought the difference might only be apparent because my solid geometry skills are arguably not great; however, there is an essential difference: successive addition of $\text{d}\hat{\phi}$ traces a latitude. But movement in " the perpendicular direction" traces a great circle! I would be grateful if anybody could resolve the apparent disparity.
[note: some errors due to negligence were corrected]