james- that's a fascinating account, & should be required-reading for anyone attempting repair on an elderly synth, especially if the schematics are not available.
I did a double-take at the first image, as I own an SH3a.... this ace tone is certainly very similar in layout & perhaps concept to the early roland. my SH3a dates from 1974-5.
roland's first synths were the SH1000 & SH2000; both of these feature the same filter circuit as the original SH3, which got roland into some legal hot water, hence the "a".... but you can read about that elsewhere.
I have three of the SH1000; two working & one with some similar faults to your half-way-fixed ace tone. if they are indeed the product of the same designer, or his proteges, then you may benefit from studying the roland circuits, at least for their principles.
(in the same way as the realistic MG-1 is similar to the moog rogue, though in fact the MG-1 came first).
all three of these early roland SH synths use a high-frequency VCO & divide-down circuitry, an idea that was common practice in electronic organs but (from a moog/arp perspective anyway) rather unusual in synths. this is why the SH synths, & your ace tone, have a single VCO with multiple "taps" in different octaves.
(I bet you can combine the octaves by simply forcing down two or more of those switches....)
these rolands also do some waveform shaping; the SH1000 & SH2000 have some rather interesting interpretations of "real" instruments, conjured by combinations of these octaves & waveforms. all three of them have two LFOs.
I have noticed also that these SH synths track at somewhat more than 1V/octave. it varies... typically 1.2-1.5. I think the problem was in getting the maximum working linear range from a particular selection of components; the tuning procedure for the SH1000 is quite involved, & the carefully chosen transistors are in a metal enclosure for thermal stability. since none of them have cv/gate interfaces as standard, it wasn't an issue at the time. it will be when I hook up my 303 to one of the SH1000s.... :-)
the missing components in your ace tone are almost certainly something to do with the "sample-and-hold" circuit used to register the keyboard voltage & store it for the oscillator. in the roland schematic, I can see matched pairs of FETs, quite an exotic component in the early 70s, & that may be the mystery 7-legged part you're missing. it may have failed. the SH1000 I have with multiple faults has certainly failed here. it also has one dead LFO...
I think the operating principle of these FETs was to present a high-impedance "barrier" to the capacitor, to prevent it discharging once the voltage was stored. so as long as the CV input of the oscillator itself isn't causing the voltage to wander, you might be able to get this part functional again with a simple switch, operated by the keyboard trigger, that allows key volts onto the capacitor as the key is pressed, then disconnects so that the only place for the capacitor to discharge is the CV input of the VCO, which should take a long time...
if I were you, I'd look at stripping out anything that remains of the circuitry between the keyboard & the VCO, & replacing it with something more "modern", e.g. the design that's used in a rogue or a prodigy. it would work just as well even if the scaling is different, & you would be able to get the components you'd need to build a copy of the moog circuit reasonably easily.
hth-
duncan.