First, ( or have you checked?) the valve that is clacking on start may not be the same one that is burned.

Second, Valve lifters are flakey devices. You may have one a little sticky or gummed up, and if it IS the one that's burned, it just takes a little bit to pump up, take up the slack, and quiet down. The same thing can happen, whether the valve is burned, or not.

Also, if the engine has ever had a valve job, the valve stem height's may not have been properly dealt with by the head machinist, and one valve might be a little short, to start with.

Same thing is true of the cam, lifter, and pushrod. You might have one lobe that is starting to wear, samo for the pushrod and lifter(s)

If the noisy lifter is on the burned (or bent!!) valve, it may just be starting to 'burn'. Also, there are situations where valves get carbon or other "junk" caught in the valve. This could even be a piece of stuff that went down the intake!!