Why do CRTs have 3 electron guns?
The first color TVs were built entirely from analog components. It would have been extremely difficult to sequence three colors through a single electron gun with the technology available at the time.

Also, the separate guns allow the separate excitation of the corresponding sets of phosphor dots through the shadow mask precisely BECAUSE they are in physically distinct locations. It is the distinct angle of arrival that makes sure that each electron beam excites only the color that it is supposed to.

Remember, the phosphor dots are MUCH smaller than the diameter of the electron beam when it reaches the screen. If you had a single electron gun and no shadow mask, the phosphor dots would have to be somewhat larger than the beam diameter in order to prevent "bleeding" among the colors, which would make them objectionably large ("grainy") when viewed.



That said, there was at least one experimental design that did use a single gun and time-division multiplexing of the colors. It used vertical stripes of phosphor, with an extra inward-facing stripe included in each group. This inward facing stripe produced bursts of light that were caught by a photomultiplier built into the CRT, and these pulses were used to keep the color multiplexing circuit in sync with the actual beam position.

Needless to say, it never caught on.