People-fixing is pointless, and generally people-fixers do it because they want to feel special, like they were the only ones able to "fix" the problem/person. So it's actually all about them rather than a selfless act. People need to take responsibility for fixing themselves - which you're on the way to doing, OP, so I take my hat off to you for that. (And I'm sorry if my previous post came across as rather terse.)
I don't think your DP sounds like he's compatible for you. The Kevin the Teenager stuff is passive aggressive, and certainly not mature. Blatantly stealing this from Baggage Reclaim - but:
"Have low self esteem —> Attach yourself to someone who will need to make you the exception to their track record of not being a good relationship partner —> Invest yourself in this limited relationship and attempt to get a return on investment —> Hope that if you eventually get him to change and see you in the way that you want to be seen and valued that this will give you an enormous boost, and make you a valuable, validated person = value inflated."
Except if you're not getting him to change, that's not validating you, and instead is feeding your feelings of unhappiness. "If you loved me you'd change" is implied in your reply of "that's not enough" to his "but I love you". But you seem to be setting him up to fail. Can I ask, did he ask you explicitly to take on responsibility for family birthday cards etc, or are you playing the martyr card by taking it on, unasked? (If he did ask you, and you agreed, then why did you agree?) Maybe your frustration is more about a perception that your inability to change him reflects badly on you, as much as his inability to change.
What would happen if you just stopped "enabling" him and picking up that slack? What would be the consequences of that for you, vs the consequences for him? Maybe he needs to experience the consequences of forgotten birthdays etc in order to learn? Maybe he genuinely doesn't think those things are important.
Back to Baggage Reclaim: "If you value acceptance, you can’t not accept who you are as a person, or engage in repeatedly letting your partner know your dissatisfaction and making it clear that you don’t accept them because you want them to change…while at the same time wondering why they don’t accept you as you are."
"...Ask yourself: What are my values? Then look at your relationships and see if they match up, and if they don’t, ask yourself why they don’t, and you start to have the answer to why your relationships are not working."