Hi I'm Brian and I'm here to help!
Normally yes, the existing start cap and relay are removed when a two wire hard start/relay kit are installed. However, many times when I've had a really stubbornly locked up compressor I've disconnected the existing start cap by removing one of the wires from the terminals on the existing start cap itself while leaving everything else still hooked up. I then added the two wire hard start kit onto the two run capacitor terminals as if there was no existing start assist to begin with. If the compressor fired up at that point, I would remove all of the existing start kit accessories and leave the new hard start installed.
This is where the however above comes in. If the compressor still wouldn't start up with a new hard start kit, I would reattach the wire to the existing cap as a last ditch effort to get the compressor fired up. If the compressor then fired up with both start assists in the circuit, I would inform the customer that I got the compressor to run but chances were good it was very close to permanent failure.
It's very possible that's what your tech did. This will put a REALLY hot start onto a compressor AND the contactor. The heavy carbon tracing on the contactor is not surprising. The breaker tripping after the unit runs for a few minutes can mean either the contactor points have degraded to the point of not passing full voltage to the compressor OR it could indicate your compressor is seizing while running. It sounds like your compressor is at its end stage. You could try to have the contactor replaced with a heavier duty, say 40 amp, contactor but that could be throwing good money after bad.
Let me know if you have further questions on this subject, use reply. Otherwise, please remember to rate before leaving (3 stars or higher please!). Bonuses are always appreciated!