i did some field testing last night...
It is very difficult to keep every component chem sprayed. The moment you are away for a rebuild, a component will come off the spray. That said, I think experienced pro engineers (like the most people on this thread) can keep the spray going.
Extinguisher is very easy to use. For a newbie player like me, the Extinguisher is more effective. And on a high engineering-difficulty ship like the mobula, extinguisher is probably more applicable. (but what is a mobula doing getting flamed? Dunno, but we came out of that fight alive)
I do bring extinguisher on a mobula because it's too large and you need to shoot to often for chem cycles to be viable, though if you're a brawler mob and the enemies have flamers, you're still probably better off with chem.
Sometimes you have to prioritize your rebuilds and chem sprays. Need to rebuild an engine but afraid you'll lose chem on the hull? Rebuild it halfway, do a quick spray run, then go finish the rebuild. Need to rebuild the hull but afraid the engines will lose their chem? Screw the engines, camp the hull, let them catch on fire -- it'll take a long time for them to burn out anyhow. Good engineering is all about prioritizing and time management. With time you'll instinctively know where to run and whether to chem, repair, or rebuild.
But until fire gets nerfed.... chem spray, chem spray, chem spray!