This evening, I did some patent searching, and I found the RCA patents that explain the design and operational theory behind how focus coils work. There are two main patents corresponding to
the focus coil design in RCA's early color TVs and
the coil design used in RCA CTC-15 and later chassis designs. The second patent describes a refinement to the coil design that allows it to adjust for a larger range of voltages than the earlier coil design.
Vintage TV focus coils look like an early form of a buck boost transformer. Just like early television gave rise to the first flyback power supply (the first switching power supply), it looks like early television also gave rise to the first buck boost transformer. Someone needs to make a list of everything in modern electronic devices today that was born in early television. I bet it would be surprising how much of vintage TV tech is still alive today in electronic devices manufactured in 2022.
The patents mention the advantage of a focus coil versus a potentiometer is a greater power efficiency, which is important for non-solid state color TVs, which pushed the limits of power output for a horizontal output stage with no transistors. So while a pot could be used in place of a focus coil, it would place a greater power drain on the horizontal output tube and the flyback transformer. Both of these components are pretty much running at their limit in non-solid state color TVs, so I think it is best to stick to a focus coil after reading the patents.
My TV uses the second (improved) type of focus coil, and I think an exact match focus coil isn't required. I just need to replace with a coil that can be adjusted to a voltage that is close to the optimal point in the OEM coil.