Yes. Usually they get a new handler. And Servitors are citizens, and killing a citizen for no good reason will not bode well for the one who did it, even if it is a Luminous Knight. And if they did have a good reason, well the handler will be blamed for not handling it.

Of course, there are cases where the handler and anyone in the general vicinity is killed, but those are pretty damn rare.

[@Cyrania]For narrative reasons, you cannot start as Servitor. You can mention that in a bio, but that would simply delay things for others.