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How did you flushed the unit? did you pump vinegar through it?
I can't think of anything off hand that fits all your symptoms. I'll get back to you if I do. In the mean time I'm going to open this backup to my colleagues so one of them can help. Please don't respond (even to acknowledge this post)until you get a response from us. If you respond first it will only delay your answer.
This is either a bad valve seat in the water heater flow controls, or more likely a bad valve seat at the faucet you are drawing hot water from.
In order to determine which do your pulse testing at various different faucets in the house... if they all pulse the problem is with a valve seat in the hot water shut off valve near the water heater, or inside the water heater.
You can operate any of the shut off valves outside of the water heater in the hot water line slightly while you feel the pulsing, if it stops with a slight adjustment, that is the bad valve seat most likely.
Tell me what you find, we can go from there without a time limit.