Why is lithium bromide used in absorption refrigerators?
Your summary is correct; $\ce{NaCl}$ is more soluble than $\ce{LiCl}$ or $\ce{CaCl2}$, for that matter, but both $\ce{LiCl}$ or $\ce{CaCl2}$ are far  more hygroscopic. The explanation of hygroscopicity involves various mechanisms, including adsorption (surface forces) and absorption (requiring transport to the interior of the substance) due to hydrogen bonding, coordinate covalent bonding etc.

So these are distinct, though related, properties: solubility (limit on how much solute the solvent holds) and hygroscopicity (how avidly the solute attaches to the initial molecules of solvent).

For that reason, $\ce{CaCl2}$ is used to absorb water in damp closets, though it will eventually deliquesce into a sodden mess.

BTW, Isaac Asimov wrote of a compound so hygroscopic that it dissolved before it hit water: The Endochronic Properties of Resublimated Thiotimoline, often cited.