The "man-ko-" term seems to confirm why, as I mentioned at the top, mancar in Portuguese came to mean to limp. To say someone is "manco" is effectively to say they're maimed in some way.

"Sleeve" in Portuguese is "manga", which in turn is also the word for mango, the fruit. The Latin "manica" meant "sleeve", while the name of the fruit originally comes from an Indian language, into English, via the Portuguese.

Etymology is quite the rabbit hole!