I am NOT talking about the main electron gun. I am talking about the deflection plates.
Just to be perfectly clear, a CRT has three active components:
1. Electron gun to create a beam of electrons aimed in the direction of the screen.
2. Vertical deflection to move the beam up and down.
3. Horizontal deflection to move the beam side to side.
Here, you are talking about #1, and I'm talking about #2/#3.
If initially the position of the beam is in the center of the path , a charged #2 and #3 will change the direction and in a smaller amount the velocity of the electrons. The energy to change the velocity comes from the energy in the electric field, not from the plates.
You can say that this energy is energy of the electric field or that it is potential energy of the electrons in the electron beam. Both descriptions are accurate and give the same result. Just remember to not count it twice.
Now the question is where does the energy from the field or the potential energy of the electrons came from. The answer is that the repulsion has a small but not 0 effect outside the main part of the capacitor and this creates an electric field outside. To align the beam at the center you must overcome this small force, so you must add some energy to the electron (in this case it's easy to think in the potential energy of the electron). Perhaps the comment from ballenarosada is more clear https://news.ycombinator.com/item?id=17432975
In a normal CRT the electron bean is created from a high voltage source so it moves fast, and the electrons initially move inside wires, so you usually ignore the small potential field of the capacitor far away. If the charges move slowly and freely then you must consider the effect of the electric field outside the capacitor.
Also, when the electrons are extracted from the cathode it gets a small positive charge, but this charge also circulate and is not accumulate. Moreover, it's the charge that eventually neutralize the charge of the electrons.
I think there are no special wires to make this happen, the current I very small and the charge is redistributed by the supporting parts.
A good comparison would be to use a CRT where the screen is split in two perfectly isolated parts. After a while the half you are sending the electrons two become too charged and it deflects the electrons.