Just musing here... If partners are in a same-sex relationship, is it still the responsibility of the partner rather than the son to organise birthday cards and keeping in touch? Or does this literally only apply when there is a woman who can be delegated to?
And for MILs who know that it is their DILs who organise birthday cards/presents rather than their sons, who do you thank?
I mean, when you ring up your sons, do you suspend reality sufficiently to say to your son, "Thank you, sweetheart, for the lovely present and card that you sent", even though you know he probably doesn't have a clue what was sent and the nice jumper or socks or whatever were almost certainly chosen, wrapped and posted by your DIL?
Or do you give credit where credit is due and say, "Please say thank you to Susan for sending a gift on your behalf - she has great taste and I really like the socks"?
In which case, why put everyone through this rigmarole? Why not just tell the DIL that you know your son doesn't care a jot about your birthday and why not just exchange gifts between the two of you (or agree not to exchange them)? And why give credit to a grown man who has done absolutely nothing? It's like sending a small child to school with a gift for his teacher, and while the teacher says, "Thank you Max", she knows that the gift is really from the parents.
Except men aren't small children. And letting them off the hook in this way and allowing them to take credit for other people's efforts does no one any favours. It harms women on a much broader level than just in family situations. There is a well-documented professional phenomenon of men disproportionately taking credit for women's work in the workplace. Why do they do so? My guess is because they feel entitled to have others (women) facilitate them and make them look good. It's an extension of these domestic attitudes.
If I knew it was my DIL organising all the gifts, I'd tell her to feel welcome to stop but, if she wanted to continue, please send the gifts in her own name only and I'd cherish them as being from her and reciprocate accordingly.