Thanks everyone. Life happened and the project got benched for a while, but the carbon pile assembly has now been rebuilt. I individually sanded both sides of almost 120 little carbon discs and wiped them down with isopropyl. Tedious.
Now to move on to ordering a capacitor, but first a few more questions...
Capacitor across the motor plus inductors is to lower EMI from the brushes arcing, so you can sew and listen to the radio. I see that in the SCR/TRIAC foot pedals.
I looked into it and it seems my model (Singer "Touch & Sew" 603) doesn't have any capacitor(s) at the motor. Not saying it wouldn't benefit from it, but I'm probably going to let that ride for the moment, unless someone contends there's an urgent need to have one in there. (brush/commutator life?)
I would use a vanilla X-capacitor 0.1uF at least 275VAC.
How did you select that value? Was there reason beyond the fact that that's what was in the pedal in the picture you shared? It's a UK 240V pedal. Will 0.1uF still be appropriate for a USA 120V pedal? I'm not versed in this EMI suppression business. Any explanation or links to a crash course would be helpful.
Also, any recommendations on what filters to use to narrow down my DigiKey results? "EMI supression" is not an option

and I'm not sure what I should be looking for. I very well may buy the exact one you linked to, but I'd like to get an idea of where it resides in the broader landscape of options.
But see pics Singer later used two capacitors, 0.1uF and 0.0047uF
I stared at that picture and other examples of the same style pedal online for some time. There is a pair of contacts that are normally open which keep the controller open-circuit when the pedal is released. The first event as the pedal is depressed is the closing of the contacts which completes the circuit through the carbon pile. The 4.7 nF capacitor is wired across these contacts. My pedal is a different style and it has no analog to the 4.7 nF cap. In my pedal, the carbon pile is actually split into 2 banks and the circuit is closed/broken between the two banks. I'd have to connect the cap to a moving plunger on one end and directly to graphite on the other, so that's a no-go. I wonder if the design was changed to accommodate the second capacitor for further arc suppression.
DON'T buy the Rifa PME271's (still for sale?!)
they fail bad, OMG they are in sewing machines too.
So, the results for a "singer sewing machine capacitor" search on ebay are almost entirely RIFA PME271 caps.

I'm really glad I decided to post my questions about this here on EEVB before I bought anything! Thanks for the warning. Is there a thread about these caps' reputation? I did a search, but there were hundreds of results.