Would you just accept that your child will need to spend a portion of their birthday at home in order to receive gifts from some family members?
Yes - at some point close to their birthday, it’s polite and nice to make time to see people who love you enough to give you presents. Frankly, you should want to see these people. Not instead of other friends, but as well as them.
Would you mention to the family members that you would appreciate them being a little more organised in future so that you aren't then trying to coordinate visitors on special occasions?
You are the one who needs to get more organised, and let everyone know in advance that a particular time is set aside for family birthday visits. Basically, plan a small family party / drop in time. This is very normal.
I'm currently leaning towards fobbing them all off on my child's actual birthday (we will be busy it's not a lie) and giving them the same time slot the day after
This is exactly what most people do - set a time aside close to birthday time when it’s convenient for you, and let everyone know that’s when they can drop in.
so that they hopefully feel slightly uncomfortable and organise themselves a bit better next year.
If there are far more people than you can host at once, plan two short events. It’s a bit rude to deliberately plan for them to be uncomfortable. But not rude to suggest lots come together.
You asked about motivation - if I love someone enough to give them a present, I also would like to see them in person to wish them a happy birthday. I do like seeing them open presents where possible, as I would like to see if they liked what I’d chosen, share in enjoyment. I like seeing the people who give me presents because I prefer thanking them in person, and I like those people.
If you view the whole thing as transactional, which you seem to do, it doesn’t make sense because a present is just a necessary social convention.
It’s possible your eldest scoffed at her grandparents when she was younger partly because you did. Now she’s more mature she realises it’s nice to see people you love.
As the adult you are responsible for making time for people to see your child, but by taking control of this yourself you will stop people staying for hours and taking up too much time overall, which isn’t great for your child. Stop thinking everyone should be like you and drop things off in advance, and plan around the reality.