Ok, some time to respond.
Much of this is from my thoughts and notes about modifying my own Goodmans units to clean up the crossovers which are poorly designed and always the weak point in the speakers.
Missing Crossover Filters (Beancounter Problem)
In the late 1960s and early 1970s market forces put great pressure on speaker manufacturers and corners, wherever possible, were cut to reduce costs. Many manufacturers cut corners, and Goodmans was not the only one. (HiFi hobbyists always malign accountants, but the fact is that management had to direct the engineers to hack up the crossovers and tell manufacturing what parts to purchase, so the fault really therein lies.)
The biggest savings came from running drivers full range, using only a first-order filter (capacitor) to control the low-end response and relying on the surround to mechanically limit the driver instead of adding inductors to the woofer and, if a three-way, the midrange. The effects will be deleterious on the sound:
Three-Way
Omitting the inductor for the woofer causes it to overlap with the midrange
Omitting an inductor for the midrange causes it to overlap with the tweeter
Two-Way
Omitting the inductor for the full-range (i.e. woofer/midrange) causes it to overlap with the tweeter.
Again, while beneficial for the bottom line, the sonics suffer. Most consumers could likely not hear the difference given the amplifier and source material (tape or LP) of the day and likely would not care.
For either configuration the resulting sound is muddier than with a full crossover. I have several Goodmans models which omitted the full crossover as a cost-savings measure, and also have a number of inexpensive speakers likely OEM'd by Coral which also omitted the inductors for the woofer and midrange.
The argument could be made that the fall-off in response of the whizzer cone was intentionally being boosted by a tweeter, but that's really not a good argument. The treble would be far too loud, since a full-range + tweeter configuration would be sufficient in treble and balanced, as would a woofer + midrange + tweeter. The tweeter, in face, often requires a little bit of attenuation. It's just a rolloff issue saving money at the expense of distortion.
Remedying the driver overlap will greatly clean up the sound.
Existing Crossover Point
I do not have the datasheet for the Twinaxiom 200c (or Axiom 200c) driver, but I do have an ad which suggests pairing with a tweeter; see the attached file.
The Axiom 201 — as far as I know, the Twinaxiom 201 is the American version of the British Axiom 201 — is a 15 Ω driver which was the standard impedance for Goodmans drivers of that ilk.
15 Ω Driver
Assuming a 15 Ω driver, the capacitor value works out to:
Given:
C = 4 µF
R = 15 Ω
Then:
fc = 2,653 Hz ≈ 2,600 Hz
I suspect the target goal was actually 2,500 Hz, but the 4 µF PIO capacitor was a stock value for motor-run capacitors and the unit was already in use by Goodmans for other lines and thus had benefits for simplifying stocking and volume discount. Consumer wouldn't care.
16 Ω Driver
If the speaker is 16 Ω, the numbers slightly downward shift:
Given:
C = 4 µF
R = 16 Ω
Then:
fc = 2,487 Hz ≈ 2,500 Hz
Which is spot-on for 2,500 Hz.
Again, I suspect the motor-run capacitor was simply handy and inexpensive. The exact value for 2,500 Hz would be 4.24 µ which would have required a second capacitor to approximate. Life for crossover designers was simpler in those days.
Proposed Crossover Modification
Full-Range Inductor
What I would suggest, assuming you're willing to put a small money into the project, is add an inductor to better blend the full-range with the tweeter. That will remove some comb artifacts from multiple drivers operating in the same space without being within 1/4 wavelength and the distortion from higher frequencies in the cone.
You could keep the existing 2,650 point or downwards move it to 2,500. It really doesn't matter as long as both the capacitor and inductor match.
I suggest an air-core inductor which is not that expensive. To cross at 2,500 Hz at 15 Ω would be 0.95 mH. At 16 Ω it would be 1 mH. So not critical.
That would give you a smooth first-order blend between the full-range and the tweeter.
Inductor Value
For 2,500 Hz the inductor value would be 0.95 mH,
fc = 2,516 Hz.
A 1 mH would be 2,387 Hz which requires a turn or so to be removed. Without access to an inductance meter this change is impossible.
Parts express has this for $10 to $26 depending upon gauge and construction:
So for $10 per speaker you will remove the overlap.
Capacitor
The PIO capacitor is degraded and muffling the tweeter, making it sound even worse than it would with overlap alone.
I therefore suggest adding a polypropylene capacitor suitable bypassed using 0.1 µF and 0.01 µF to improve high-frequency performance.
Crossing at 2,500 Hz would be 4.24 µF which can be built using either:
C = 4 µF + 0.15 µF + 0.01 µF
fc = 2,550 Hz
C = 4 µF + 0.15 µF + 0.1 µF + 0.01 µF
fc = 2,490 Hz
Either one is close enough given the tolerances and fiction of nominal impedance. It's not critical.
Proposed Extension: Super-Tweeter
Paper-cone tweeters have poor high-frequency performance, and the Goodmans are no different. I'd keep that because it will match the characteristics of the full-range, but I would not expect the high-frequency performance to be good, and the tweeter will distort with higher frequencies. Just a fact of life with paper-cone tweeters.
If you like the sound, you might consider adding a super-tweeter as has been done for the Bozaks. My design crosses the Bozak tweeter, an aluminum cone design, at 8,000 Hz, which is high enough that the point is not taken in a region in which human hearing is highly acute, yet low enough that the distortion from the mechanical errors is minimized. A modern soft down then runs from 8,000 Hz to 24,000 Hz, limited to prevent ultrasonic oscillation from damaging the tweeter. You could try that, as the paper-cone should be fine at that frequency, operating without distortion.
The cost of adding (1) an inductor to the existing tweeter to form a band-pass and (2) constructing a high-frequency band-pass for the super-tweeter is small. Parts Express lists a 0.3 mH inductor for 8,000 Hz is $3, and the 0.1 mH for 24,000 Hz is $4:
The cost is basically stocking and putting it in a box.
The main cost here is the soft-dome tweeter.