What breaks first
Updated
The kettle
A thick sauce scorches on a surface that is hotter than the product, and the bigger the batch the worse the ratio of surface to the volume you can keep moving. A stockpot that behaved at two gallons will burn at eight, and the first scaling expense is almost always a vessel with better heat distribution rather than more capacity.
The fill
The product has to go in hot, which means the whole batch has to stay hot while it is being filled. At small volume that is free; at larger volume the tail of the batch is cooling in the kettle while you work. Either the fill gets faster or the batch gets held, and holding a thick sauce for an extra hour is its own cost in color and in scorch risk.
The cooling
Jars that are not cooled promptly keep cooking themselves, and a case of hot jars stacked tight cools from the outside in. The failure shows up as darkening and flat flavor rather than as a safety event, so it is easy to miss and easy to blame on the recipe.
What does not break
Prep labor scales linearly and predictably, which is why makers consistently overestimate it and underestimate the three above. Chopping twice as much takes twice as long and nothing surprising happens. The surprises are all in the thermal part of the day.
If you are planning one investment, make it the kettle. It is the only one of the three that also makes the product better at the volume you are already at.