"efficient" and "clean" aren't the same thing, and they never have been.
Recycling lead-acid batteries is extremely efficient. Nearly the entire battery by mass is recovered.
But, it also causes severe lead pollution around recycling sites. Lead acid battery recycling is one of the leading causes of lead poisoning around the world [1]. Estimations vary, but all generally agree that millions of human-years of life have been lost due to lead pollution caused specifically by lead-acid battery recycling. [2]
[1]: https://link.springer.com/chapter/10.1007/978-0-387-77030-7_...
[2]: https://pmc.ncbi.nlm.nih.gov/articles/PMC5990833/
Returning to the original point, recycling anything involving heavy metals is extremely difficult to do without poisoning people. If we can't avoid it with one of the simplest, dumbest battery technologies in regular use today, I don't see how we're going to avoid it with a battery technology involving heavy metal nanoparticles.
My reading of both those reports isn't that lead can't be safely recycled without contamination, but rather that countries with low regulations and oversight aren't recycling lead batteries in a safe manor.
In fact, the second link is more about the problem with using smelting to recycle lead. That requires a lot of power and thus emits a lot of CO2.
Is it the case that lead acid batteries are being primarily recycled through exports?