Tesla coil help - Spark gap not arcing despite having 12 kV
A car ignition coil is far too small output current to be loaded by a 13nF MMC. As you mention, even a multimeter drags the output down.

Usually we use a Neon Sign Transformer (NST) with 10s of mA output current in this sort of setup. I notice you started with NST, which explains a lot! If you hired a U-Haul trailer, then found your car didn't have a tow-ball, would you tie it with string to your bicycle? You need a car for the trailer, you need an NST for 13nF. You have the reasons written on your diagram, coil output \$600\mu A\$.

If you want to continue with an iggy coil, then suggest you rectify the output so it can take its time to charge the capacitor to a high enough voltage for a decent spark. DC coils can work, though a rotary spark gap (RSG) is often necessary for extinguishing the arc after firing. Otherwise, tune the MMC down to work with the iggy coil. Either way, you need to change the design.

Unfortunately GFI-less NST's are becoming rarer, they're not being made these days. But that's what you need to make an unfussy TC that will just work. 

SO is a very general forum, Tesla coiling has a very small active community. You would be far better off asking these questions on a dedicated TC forum, you'd get a much better signal to noise ratio. Start with the pupman lists (archive) or 4HV (active) for instance, where you can find the community that has the stuff, and the domain knowledge.