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, that explains a lot! It's like you hired a U-Haul trailer, found your car didn't have a tow-ball, so tied it with string to your bicycle. Untie the trailer, and the bike works.

If you want to continue with an iggy coil, then suggest you rectify the output to let it charge up 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.

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. There are some specialist forums for TC makers, see the pupman lists for instance, where you can find the community that has the stuff.