Rafter load capacity
Your roofers are correct in adding plywood or osb board I order to maintain the roofing warranty. If there is a “blow off” the manufacturer will not honor the warranty if the nails are not installed correctly, including the minimum required thickness of the roof sheathing, which is 1/2”. (1x6 boards is not acceptable because of the number of gaps between boards means the nails could miss the lumber and be installed in the gaps, and thus no withdrawal holding power.)

We design for the exceptional not the standard. So, your occasional 24” deep snow will translate to 30 - 35 lbs. per square foot Live Load plus about 15 lbs. per square foot Dead Load (roof sheathing, roofing, etc.). 

Your 2x4 roof joists (even the older larger ones like you have) will only span about 9’, depending on the species and grade. However, the occasional diagonal 2x4 brace will help, but your roof joists are over stressed. 

I’d also be alarmed about the connection at the ridge. High winds could “suck” the joists off the roof with just nailing. I’d use Simpson joist clips to hold the roof together.