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Why couldn't I just send whichever other party that I'm paying to print my key gadget the final form of the encrypted bip 38 key?
You can, as far as I know from reading BIP 38, this would be just as secure.
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What's the advantage of the intermediary code?
Sample uses cases are in the BIP:
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User story: (EC-multiplied keys) As a user of physical bitcoins, I would like a third party to be able to create password-protected Bitcoin private keys for me, without them knowing the password, so I can benefit from the physical bitcoin without the issuer having access to the private key. I would like to be able to choose a password whose minimum length and required format does not preclude me from memorizing it or engraving it on my physical bitcoin, without exposing me to an undue risk of password cracking and/or theft by the manufacturer of the item.
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User story: (EC multiplied keys) As a user of paper wallets, I would like the ability to generate a large number of Bitcoin addresses protected by the same password, while enjoying a high degree of security (highly expensive scrypt parameters), but without having to incur the scrypt delay for each address I generate.
Basically, it allows you to easily generate many keys from the same intermediary code (thus all with the same password).