My brain read the OP and thought of two different scenarios. One good and one bad.

Good Scenario: Your friends invite you to birthdays because they happen on the same day each year. If their "spur of the moment" claim is true, that makes sense seeing as every birthday is expected a year in advance... until someone dies lol.
Bad Scenario: Your friends only invite you to birthday events because they want the present. That's it. They do not invite you to other things on purpose, just as you feared.

I don't know you and your friends so I don't know what's going on here. Whatever it is, I hope you get to the bottom of it! Friends are very important to mental health and the "right" friends make all the difference. For instance, there seems to be some trust issues going on here and you should look into why. Like, why don't you believe them when they say events were "spur of the moment" and why does it bother you when they do things without you if you're busy with your family? The aforementioned questions aren't me attacking you, btw - just questions I wanna know if you've asked yourself.