Recycling steel as steel is a pretty widespread thing to do, and steel accounts for almost 90% of manganese consumption.
If you want manganese for some non-steel purpose, A36 structural steel (like girders) is 1% manganese, which admittedly is less than ideal; manganese nodules are up to 30% manganese, and in the US people mine manganese ores of less than 5% manganese content. (The ores sell for about US$7 per tonne, and given that manganese is the 12th most abundant element in the Earth's crust, they might always be cheap.) But if you're looking for ores, well, discarded non-rechargeable alkaline batteries are more than 30% manganese, so you're probably better off digging through landfills than diving to the bottom of the sea.
I think X-ray fluorescence scanning is a thing that minimills have been doing for decades.
Overall I think we should expect recycling to be a major industry. Your point that lithium batteries are a far higher grade source of lithium than ores or brines is a good one, and applicable to many products, though I'm not sure if it's literally true in this case. Given lithium-ion batteries' energy density of about 500 kJ/kg as against the 13901 coulombs per gram of the lithium ions themselves, working out to 51 MJ/kg Li at 3.7 volts, the batteries can't be much more than about 1% lithium, and I think there actually are Bolivian brines that are richer than that, certainly if you omit the water. But for many resources, deposits found in landfills will be the richest available terrestrial resource for a long time.
For many minerals the USGS Yearbooks from four years ago give a more complete picture than the MCS, which are indeed great: https://www.usgs.gov/centers/national-minerals-information-c...
The Murphy et al. article points out that we can get through the next 1000+ years at current growth rates without even expanding beyond the solar system or unlocking new sources of energy such as Hawking radiation, though for some reason it's pessimistic about our chances of doing so.