Hi beachpony
That "button" is the UNWORN metal that DID NOT wear away because it aligned with the hole in the shaft.
That button is showing you how much metal is actually worn away from the entire lower half of the rocker arm bore around it.
Your old shafts are SEVERELY worn, and look like they have been getting NO oil for some time.
If the head you are reconditioning is from an engine you have, you will need to investigate the reason why no oil has been reaching the rocker arms.
So, hate to be the bearer of bad news but...
Unfortunately, if I were you, I'd be looking for a new set of rocker arms as well.
PS
If you find a set of used but still usable rocker arms,
do NOT recondition the end which contacts the valve with a bench grinder.
You will ruin them.
Take them to a shop with the proper equipment to do the job.
Last edited by Stovblt; 02/24/24 08:34 PM.