Why, viletim?
Well, to clarify, there's nothing inherently wrong with capacitive coupling, it just needs to be done properly. Coupling through a capacitor removes a some vital information from the video signal, the black level. This is a fixed voltage that is declared to be "black". Any voltage above this level is active video information. If this information is removed it must be put back a some stage before it gets to the CRT cathodes. This RGB amplifier circuit doesn't so the monitor must do this.
Most old monitors aren't prepared for AC coupled video and won't display it properly. At least that's what I believe. I downloaded 10 random monitor manuals from
http://www.mikesarcade.com/arcade/monitors.html (when the links still worked) and two of them had internal video clamping circuits (which would restore the black level based on the knowledge that the video signal is "black" when the sync pulse is active). From what I've heard from various people on on the internet, it's clear that most new (less than 10 years old) monitor chassis don't care to much about this.
I leared from the other thread all about the PC Enginge's wacky video. When using a transistor amplifier there's simply no choice but to couple the signal with a cap. With the addition of a single transistor per amp (or just one cmos switch IC like the 4066) the video can be clamped to the sync pulse and everyone would be happy. Another benefit of this would be an increased output swing (bigger signal on output).
And finally there shouldn't be any need for pots on the output. The maximum voltage swing on the output of this amp is already less than a typical arcade board. Why would anyone want to reduce it? Just another thing to be mis-adjusted.
(you did ask)

My guess on the microcontroller is that it controls the coin timer related stuff (like switch debouncing -- very important for a coin input!) and very little else.