Should I replace a single powered roof vent with ridge vent according to my roofer's advice?

In my personal experience, a "real vent" on top of the roof (cupola, turbine, etc) is far more effective than continuous ridge vent, and I will in fact be retrofitting a cupola onto a roof with continuous ridge vent for exactly that reason. The space feels nearly unvented, despite "being vented" as the air movement through the ridge vent (with plenty of venting at the eaves) is pathetic.

What you have there appears to be a "mushroom fan vent" rather than a turbine vent (i.e it only really moves air when powered up, while a turbine vent as I understand the term vents passively all the time and more actively whenever the wind blows.) That can be useful for increasing air movement in the space when needed, and should not be particularly prone to leak if it's in good shape. It would not meet code requiring passive ventilation, though.

I would strongly suggest reading the actual code applicable to your local area to see what it actually calls for, rather than accepting that it specifies "ridge vent" if, in fact, it simply requires passive vents but does not specify how that is to be done (which would be more likely, in my experience.)

I would keep the ridge vent and add the powered attic roof fan like one of your pictures shows. Done right (proper attention to flashing), the fan will not leak. And the powered fan is much more effective at clearing hot hot attic air than any passive system.

And do call the HoCo inspector's office. When I lived in HoCo, I found them to be very friendly and helpful. They want to help you do the job right.

EDIT 1 - Replacing the roof mounted fan

I've found the fan motors and mounting brackets to be fairly universal. I had the motor go out in my roof mount fan, and was able to replace it with a motor from another unit. Yes, I had to buy the entire unit, but at least I didn't have to remove & replace the fan housing itself, which was flashed and water-tight to the roof.

while you are trouble-shooting the paint-blocked soffits, you might check that they are not blocked inside your attic, especially if you have blown-in insulation. Check that you have "rafter vents" installed which provide for unobstructed ventilation from the soffits.