Why do non-SegWit clients send witness data in the first place? If SegWit leaves out witness data for backwards-compatibility, thus witness data isn't so important apparently, then why do clients even send that data in in the first place?

Aren't they making their transactions unnecessarily bigger and thus more expensive?
Non-segwit clients do not send witness data. They do not know what witness data is and would not know how to interpret it.

However, I think you are asking about the scriptSigs, which contain largely the same data as witnesses. ScriptSigs are actually scripts, but since the only standard scriptSigs push data to the stack, the same behavior can be done by segwit with the witness data. Non-segwit clients must transmit scriptSigs (and the scriptSigs of segwit output spends are empty) because they are part of the data that is hashed for the transaction id. Furthermore, non-segwit clients must be able to verify non-segwit transactions; they shouldn't just blindly trust that the transaction is valid. So the scriptSigs provide the information necessary for them to verify those transactions.