Why doesn’t the focusing anode diverge the electron beam again?
It does diverge it is really hard to get a bunch of electrons to stay in one place, an electron beam has an internal divergence due to self potential. If you stick two electrons close together, they will start to move away from each other, in a beam this is also the case.

It can kind of be seen in the diagram below (more in the voltage gradient shown than the electron lines), as electrons exit the cathode, they start to spread out.


Source: https://www.quora.com/How-do-anodes-focus-and-accelerate-a-beam-of-electrons

Electrons are usually more useful when they are in a tight beam, so they need to be refocused. The beam focuses to a smaller point, however, if you did not refocus it it would diverge again.