The benefits of using images to illustrate a proof are plain $-$ complicated arguments and endless strings of symbols can be very difficult to understand and pictures can help to give a picture of what's happening. With suitable illustrations you can get intuition for what's going to happen next, and so on.
However, there are a few fairly compelling reasons not to rely on pictures exclusively, a few of which I have listed below.
By using pictures we are very much confined to the so-called real world. Sure, we can get lots of intuition for low-dimensional topology by drawing tori, but this isn't going to get us far if we want to study $3$- or $4$-manifolds, let alone manifolds in higher dimensions.
As category theory has done such a good job of showing over the last half-century or so, there are deep-rooted connections between all sorts of mathematical objects and fields which are not at all obvious. If we limit our understanding of certain objects and results to what we see in pictures, then these connections are likely to go unnoticed.
Mathematics is all about abstraction. By using pictures as a means of doing maths rather than merely illustrating it, we are working exclusively in the concrete. Just because a result holds in one picture I've drawn, why must it hold in all such pictures? Can it hold in other settings? Must we draw pictures of all such settings in order to prove this result? Etc.
Using pictures to do mathematics relies heavily on the accuracy of the pictures, as illustrated in the link in your question.
It is not fair to say that using pictures is the same as geometrical thinking. It might assist geometrical thinking, and it might even form the basis of geometrical thinking, but this thinking is not the maths itself. The maths is what follows, when the intuition is translated into a formal argument.
This isn't to say that pictures shouldn't be used at all. In fact, not using illustrations at all may be almost as damaging as using them exclusively! I mentioned category theory, and I can't say how much harder it would be to understand if we didn't use commutative diagrams.