Pretty much what Lana said, though not sure she'd have been sent abroad -- too far out of the Bennets' economic and social world, I suspect, plus no one would have believed for a second that wild, bosomy, 'always likely to end up in trouble' Lydia was travelling 'for her health' or education.
I'd have thought being sent to a part of the country where she wasn't known with a cover story and respectable female relative, like Maria Bertram (after she leaves her husband for Henry Crawford, only to be spurned) might be more likely. Only there's no one obvious to go with her. Her mother and sisters can't, Mrs Gardner has small children, Aunt Phillips wouldn't.
But the outlook for the rest of the Bennets would then be very, very bleak. The other four girls wouldn't have been able to marry, and once their father dies, the Collinses inherit Longbourn, so they're homeless and have only the income from their mother's small capital. Plus they've not been bred or educated to earn a living, so even governessing, Jane Fairfax-style, seems unlikely.
My students sometimes think Lizzy is mostly just mortified that Lydia has disgraced her in front of Darcy, but in fact, as at least a couple of the girls have to marry well in order to support the others after their father dies, in fact Lydia has doomed the whole family to poverty.
If the letter about L eloping hadn't happened to arrive at the one moment when Lizzy was too shocked to hide the truth from Darcy, no happy ending would have been possible. All the romance is Austen is so underpinned with economics.