In a strict sense, the EHT does not generate the electron stream ("cathode ray") in a CRT. The electrons are "boiled off" a heated cathode and modulated and accelerated by a series of grid and anode electrodes in the neck of the CRT, conventionally called an "electron gun". In a colour CRT, each of the red, green, and blue colours might have a dedicated electron gun, which "fires" the electrons at the phosphors on the front glass of the CRT. After impact, a "cloud" of free electrons begins to gather near the front glass, and would repel new incoming electrons, until sufficient numbers had gathered to prevent any new electrons from actually reaching the phosphor.
The EHT is actually connected to a conductive layer inside the "bell" of the CRT/picture tube, and acts as a positive electrode to attract and "collect" these free electrons after they have impacted the phosphor.
The flyback circuitry is so-called because the horizontal deflection waveform that controls the horizontal sweep of the electron beams resembles a sawtooth with a long linear ramp up, and short steep ramp down. The ramp down is the period in which the electron beam is returned from the extreme right to the extreme left of the CRT/TV screen, and called the "horizontal flyback" because the electron beam "flies back" to the left in preparation to beginning another horizontal sweep of the screen. Because the flyback is a very steep ramp, it has a high di/dt. This steep ramp is the "flyback pulse" and is used to generate the EHT through a step-up "flyback transformer" and rectifier.