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When Leo Szilard conceived the idea of a nuclear chain reaction, he thought it would work with beryllium. If this really works, it means he missed by one square in the periodic table.
IIUC the reaction is

  Li_7 + n -> Li_8 -> Be_8 + e -> 2 He_4 + e
it makes sense if you have a external neutron source, but I can't imagine how they can chain the reaction because it release electrons instead of neutrons.

Also, I don't understand all the recicling Lithium angle. If this go up comertially, woudn't it be cheaper to buy Lithium from a common source instead of recicling batteries? (If recicling were cheapier, batteries factories would have been using that source.)

Yeh I don't think the plan is to chain the reaction, just have a high energy neutron generator driving the reaction
I'm not sure about the cost, but I remember that for some medical aplication there are some protons rays to neutron rays "converters". (IIRC it's a target of berilium that absorbs the proton, splits, and release a neutron. IIRC) They are easy to turn off inmediately. (IIRC)