Yes, it does take close to 10 days to get the actual Social “card” posted out to you.. HOWEVER it is considerably faster (I was told overnight and mine was ready in 3 days which was the earliest I could check): if you revisit the SS admin office a second time again after your initial application they can then release your actual SS number to you.
To start work the airline just needs the SS number, not a copy of the card so this can speed things up quite a bit.
Cheers Ralphi