The point of a Tesla coil is resonance... which means AC on the switch. Your relay was a very good choice, it is a good HV open circuit at AC (and none of the solid state choices is as good).
To simply make ONE pulse, not a whole AC train, things like an IGBT switch are good, but those ignition coils only take 4A anyhow; any transistor would do. It has to turn ON and suck
4A at (or under) 14V, then turn OFF and tolerate 500V or so.To get a clean drive, an alternative (like on the old Mark X aftermarket ignitions) is to charge up a capacitor and fire 300 VDC into the coil at the 'last minute'; that makes the coil run cooler because there's no long time duration slow current buildup.