think you might be missing a point -- expansion plug is a routine replacement part that is part of cleaning and rebuilding. The plugs are included in rebuild kits. You remove it to clean out the idle system. There is no law that says you have to do it, but you risk poor idling if you don't open it up and clean it.
With that other carb**, you should expect to clean it and put in a rebuild kit, replacing its expansion plug after cleaning under there. (As you intend, push that broken tip out from the inside of the current carb, after taking off the plug. But don't drill it out.)
**(edit.) that is, if you pick up a replacement carb....