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Need countersink help

Rigidity is playing a part in this, no question. That said, I have absolutely had the same thing happen with these 6 fluters on a Bridgeport when feeding the quill with just the hand lever. Using the feed handwheel by hand with the feed drive in neutral or locking the quill and feeding the knee up will easily solve that though. Also, modifying the cutter clearance works too.
Clamp the job. You can also use a folded piece of paper instead of rag its marginally safer. If you can't go slower switch the motor off just before you start counter sinking and feed quickly before it stops.
Over the years I've tried a number of different styles of countersinks. The most forgiving by far seem to be the single flutes from either MA Ford or KEO. They can be used on a worn-out old drill press at 300 rpm, or a solid mill at 90 rpm and give a perfectly round chatter free profile. When absolute depth is required, you can't beat a microstop.