Might be wrong (I often am!), but I do not believe the circuits of the two pedals are the same, even allowing for the tone control. I believe that Boss marketing the SD-1 as "an OD-1 with a tone control" was more along the lines of "here's another overdrive pedal but this one has a tone control."
This IMO is correct.
It's easy to forget (or just not even be aware) that tone controls were still kind of a new thing with many drive pedals back when the OD-1 came out.
In the case of the OD-1 however, there are two op amp stages - kinda different from the other op amp-based stuff that came before it. So Boss figured they would put it to use, just not in a tone shaping capacity.
...But op amps are actually pretty good for employing tone circuits around.
The way the 2nd op amp stage is configured in the OD-1 requires that some things be changed to make for (what Boss believed to be) a better tone circuit. An op amp has two inputs - an inverting and a non-inverting. I won't go into details, but suffice it to say that the 2nd stage in the OD-1 has the signal presented at the inverting input, but a tone control (with reasonable simplicity) would do better having the signal fed into the non-inverting input, so that you can adjust it (like the gain/drive of the first and primary stage) at the inverting input.
...So that requires that things be swapped at pins 2 and 3 at a minimum, which is what the "Erik's Corner" page highlights.
Aside from the tone circuit components directly tied to that second op amp stage in the SD-1, there is also a fixed lowpass filter just prior to it.
So the tone circuit in the SD-1 is a "properly engineered" addition as opposed to a "let's just graft this thing in here" kind of approach. Lots of stuff that preceded the SD-1 took that route. It was rare to see something like the Colorsound Overdriver back in '71 or so, which featured active bass and treble via a "Baxandall" tone circuit, and done with old (by today's standards) bipolar transistors.
And much of the stuff just seemed to have a lowpass filter tacked on at the tail end of the circuit that was either too mild or didn't have a good sweep. Some even affected the output if they had no recovery gain stage like the Big Muff had. The Rat had a unity gain buffer after its tone circuit, but it was more to recover signal after the clipping diodes shunted to ground. Pedals like the D+ and DOD 250 had no recovery stages (and no tone circuits, obviously).
But the OD-1 must have clearly been an instance of a design that (in the eyes of Boss) would benefit from a tone control. Using dual op amps really seemed to change things, and lots of the most popular drive boxes still seem to revolve around a dual op amp design today. Timmy, OCD, Zendrive, etc...