That's correct.
However that only refines lead, it doesn't reduce lead oxides to lead as you noted. So, in short, it doesn't work with batteries because batteries don't contain (much) lead. They contain mainly lead oxide and lead sulphate.
As I posted earlier batteries, can't simply be "melted" down to get "lead". You get lead oxide and lead sulphate, neither of which are elemental lead. Just like aluminium oxide isn't aluminium or iron oxide (rust) isn't steel/iron.
Reducing (opposite of oxidizing) using a carbon source, in a non-oxygen atmosphere, is how the oxides and sulphates are removed, leaving lead (and other metals).
There is a small scale electrolytic method of using very concentrated sodium hydroxide. It is slow and multi stage and deposits lead on a cathode. That's why it's not used commercially.
You'd already covered reduction, so I felt no need to go over it again... Reducing the lead sulfate and lead oxide to elemental lead then eletrolytic refining would be the most expedient... But, the reduction would be beyond most backyard refiner's capabilities. Done improperly, groundwater could be contaminated, one could get arsenic poisoning, which is a very painful way to die, and cleaning it out of your system is worse than dying from it... And if done wrong, the treatment for arsenic poisoning can cause your brains to come seeping out your ears... It's also black market, as no doctor with a license will do that treatment...
Get wheel weight scrap. Go to your junk yard and buy casting or machining lead scrap...
BTW, the Electrolytic refining I was talking about uses sulfamic acid.
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