To be honest, I have seen piston compressors work for years with the unloader non-functional. Our shop compressor was one.... It was an IR, Type 30, 10HP v-twin, and the flywheel would bounce back and forth when it stopped, due to the elasticity of the trapped air. You know the old saying about the children of cobblers having no shoes...well, it was one of those things I was going to "get to" when I had the chance, but as the repair of everyone else's compressor came first, getting to it took quite awhile, but when I did finally adjust the unloader, I swear that machine would sigh in relief whenever it shut off.
Bear in mind, the wiring and switchgear in our shop was of industrial quality, and sized big enough that I often tested larger machines with heavier amp draw off of that same starter. It was also a very high quality motor.
Despite all this, it was very hard on the motor, and very hard on the power bill, especially when I was running a bead blaster all day. These may be the same reasons your machine could have worked for years without a proper unloader but it is certainly not the correct way to do it. The components your sparky friend installed were probably sized for the expected load, not of the overload it is now experiencing. It is also used stuff, so who knows if it all might be a bit tired. You already did some good troubleshooting by turning the machine by hand and seeing that it turns easily, so I would suggest to you that mechanically you are ok. You just need to give the compressor the help it needs to restart by blowing down the pump and discharge line with the unloader.