No, "Qui ne l'aurait pas ?" is not appropriate here. As a native French speaker it took me some time to understand that sentence. It is true that in the locution "avoir besoin" uses the verb "avoir", and grammatically I don't think there is a problem with your sentence. But "avoir" is so tightly linked with the word "besoin" that the sentence stops making sense if you remove it.
If you want to translate the sentence more-or-less literally, you can use "Qui n'en a pas besoin ?" as cccg03 suggests in the comments. One could also say "Comme tout le monde", which has a slightly different meaning but feels a bit more idiomatic.