I would do a toy-trade rather than asking for all gifts for your DD. At Christmas, it's usually called a White Elephant Gift Exchange - you could parody that name somehow. Encourage trading out toys from their own toyboxes rather than buy new, but that might be more difficult.
Personally, I've always had plenty of success saying NO gifts. I seem to be the only person who does, though. I don't think it's an indication of my community, either - but probably more of a fear of making "Janis" mad by disobeying
I'm pretty outspoken about it. My kids actually think it's weird when they go to parties where gifts are given. (they've never been to a no-gifts party, but they certainly don't complain about my rules) Of course, the in-laws disregard our request every year.
Another reason why it may be that I'm not having issues with it is that I don't make it a "birthday party" - it's a "unicorn party" or whatever theme that my DD picks. And my DD draws the actual invitation which we hand out in person to our guests. We always have a birthday cake, we have a creative activity, and a pinata, and guests do know it's a birthday celebration, but they also know that it's not going to be your "traditional" sort of party.
It kinda sounds scroogy when I type it out, but everybody does love our parties!
p.s. my dd's get their first "real" birthday party when they turn 5. They get to invite one friend per year of age. Siblings don't count, but they are always included on the invitation. Grandparents and babysitters don't count, either. Prior to that, the date is celebrated with just immediate family and cake.