Is there a plan to remove the option to generate a payment ID not using the form of an integrated address?
If not, what are the advantages of allowing a payment ID separate from the receiving address?
There are no plans to do so at this time. If there were, it'd need to leave a large amount of time for current users of payment ids to switch to integrated addresses.
One difference between new style and old style payment ids is the size: old ones are 256 bits, while new ones are 64 bits. It is plausible that some party could find a use for 256 bits to carry information which could not be carried by 64 bits, and not care about it being unencrypted, so old style payment ids are not necessarily completely useless now that integrated addresses exist.
That said, a general shift towards integrated addresses would be likely to remove some of the pain exchanges have with people not reading instructions.
One of the biggest reasons for using integrated addresses is that exchanges no long need to require users to deal with an extra address field (the payment ID) when sending Monero. Seeing how easy it is to manage integrated addresses, being adding or retrieving the embedded payment ID from a regular address, I do say its a matter of time for it to completely replace regular payment IDs.