How do you deal with incompetence at work?

Actually over my CS career, I've demoted and in some cases fired people for that. Hell, I canned an O-5 Ops Boss on the MIL side amongst others. Yes it's a somewhat onerous process, moreso on the CS side, but it can be done. Biggest thing is to keep everything clear and the employee makes all the decisions for you by what they do and don't do. And like Gunny Anderson, I did dive into the cognac stocks every now and then. But the rewards of improved morale for everyone else who were being dragged down was worth it.

Funny, I remember a GS-12 who never got promoted pounding the table and yelling he has 32 years experience. My response was he had one year's experience 32 times.

I just muddle along and hope that I've been moved to another section before people notice. It helps to be friendly and willing to assist others. You can be stupid and nice but not stupid and a jerk. Anyway, that's how I manage incompetence at work.

The answer to your question is the same in the military and civilian world. Basic leadership and management principles apply. If you are junior to the incompetent, you perform your duties and eventually those senior to the offender will note the changes and analyze from whence they came. If senior to the offender, you school them, record all their shortcomings and their response to your efforts, then follow the protocols of the organization to have them replaced. It works. No organization wants dead weight and, if nothing else, you can make the incompetent's life so miserable they want to leave. If your tenure as their senior is so short they don't leave while you are there, at least there is a record that your replacement can build upon.

The answer is: frank and honest counseling with the intent to help the person be successful since they're already on your team. We're all working with the cards dealt to us, some people just play them differently.
The point of counseling is to improve their performance, but sometimes it just doesn't work. I've helped people find work elsewhere, sometimes they transferred before they were helped out of the GS system. Same with officers and enlisted. Honest rating that is supported by sincere counseling. When ratings are performed properly, they really do write themselves so I really like your picture up top that says their work speaks for itself. That's the best outcome.

Incompetence left unattended can be a plague. It not only impacts the individual but his or her co-workers and the overall organization output. Standards need to be clear and personnel held accountable. Use to piss me off to no end when leadership would reassign a substandard performer so he/she could be someone else's problem. Who's the incompetent one?

Throat punch? Is throat punch an option?

the one word that I never liked transfer to a very cold place

LTC (Join to see), the last incompetent person that worked for this financial services practice was fired. Happened about two months ago.