That is a feature that might give the troubles you're talking about.

The idea is that the sensor shuts off spark if there is no flame (or gas present, I can't confirm this from my searches) present.
If both burners share this same sensor which is likely and this part is failing it would shut both down.

See this thread about an owner having different symptoms with the same part.
Flame detector

I think this sensor would make more sense than the module.

Can you see anything that looks like what that link is discussing? Maybe shine that sensor up with some light sandpaper?

My thinking is that if they both share the same sensor and its on its way out (or be gunned up), it may constantly think there is no flame and therefore not spark.
Once you manually light it, it's enough to register with that sensor. At that point it doesn't have any issue allowing fuel and spark.

This may all be accomplished through the spark module - I'm not familiar enough to say.

If these are the only wires you see it's got to be controlled by the module.

If that's the case, replace the module.

Excellent! Nice work.
Thanks for the follow up!