I think there are 2 separate possible problems:
AuntyRose: If your neighbors have lots of neighbors themselves, they're probably not all seeing your requests. Some people have hundreds of neighbors. Sure, some of them may be inactive or skipping the goals, but even so, everyone who is doing the goals needs tons of parts, so logic dictates that people with lots of neighbors have a lot of requests sent their way.
But I believe players only see the 10 latest requests they get, not every request they're sent. Mainly because I personally have never had more than 10 at once, though occasionally a couple more will pop up as I answer the first 10--but I just don't think that's all of them.
All of which is to say that if you're someone who has so many neighbors that gifting, even mass gifting, is too much of a hassle, but you tell everyone you accept all requests, you may not be as good of a neighbor as you think you are. (Exception for people who have request notifications on and are super responsive to them, but I'm betting the majority of us aren't.)
But that's all in a perfect world. Possible problem 2 is that I agree requests are sometimes not being sent. I nearly never ask for more than 20 (and only when I'm not requesting tomorrow) and like pipipooh I sometimes don't see my own requests on a restaurant where my only neighbors are my other restaurants. And then they pop up 4 days later, just to mess with me.