Digitally-Controlled Analog Processing for VGA Video Signal?
Thanks to @Hearth and @JohnBirckhead in the comments, I have a theoretically workable design.
No continuously variable mixing or attenuation at ~62MHz, but enough steps to be close enough.



The symbols aren't exactly correct yet, but it shows the idea.  I'm switching from Eagle to KiCad and have yet to figure out how to make custom parts in KiCad.
The power supply also needs some attention before I spin a PCB.

The de-saturation/grayscale is a 2-bit, 4-way switch using this chip:
https://www.renesas.com/en-us/www/doc/datasheet/isl59446.pdf
It's a 3-channel, 4-way switch with a x2 buffer built-in.  That extra gain accounts for the signal loss in the impedance-matched inputs and outputs.

The color inversion is an opamp buffer that is switchable between a gain of +1 or -1 using this chip:
http://www.ti.com/lit/ds/symlink/tmux1133.pdf
It also provides (an attempt at) the proper signal blanking during the HSYNC interval by switching back to +1 during that time.

The output attenuation/dimming is a 4-bit logarithmic stepped attenuator with constant 75R output impedance and ~1dB/step, using the same 2-way switch and this calculator:
http://www.vaneijndhoven.net/jos/attenuator-calculator/index.html