I think you have enough answers to proceed. Basically you need to research part numbers, and find what off the shelf water pump assembly was re-purposed for your machine; or bore the housing to fit a new one that is otherwise a close match for the purpose (has the same diameter and length shaft extension)
If any part of that has failed, every other surface will be degraded so even if you can get it apart and replace with new loose balls, it will not likely eliminate all play, or last very long.
So, your machine is reparable. But it will take anything from a parts swap, to a slight, to a major rework of the existing casting and systems. We can repair junk machines to better than new for twice the price because we are machinists.
The title of this post is not really within the rules. (no clear explanation of the subject. Could be doll parts)
The machine under consideration does not look like one that would meet the rules.
I'm curious what you come up with, so not locking yet. Makita combo machines IMO are also junk, but the iron was fair, just really crap/less than home-owner quality drives; ostensibly marketed to "pros" (lol). So a good answer could help others. But you need to read and comply with the rules or take it to a hobbyist site.
smt