What the fuck is up with everyone being allergic to things. Food allergies were unheard of when and where I was little. I can't believe that this isn't a modern phenomenon. (Unless maybe "gluten allergy" used to mean "you will have plenty more babies to replace that one"?)
Birthdays? Go fuck yourself. My parents were awesome; I got birthday gifts till I turned 11. After that: you're old enough to save up and buy your own.
Video games? Double go fuck yourself. If everyone has a phone, a tablet, and a laptop... the last thing you need is to spend money on a video game system and games. Even with any one of those three, you can get good games for free. When I was younger, though, video games were a large reason to start programming - it used to take some skill to play them, and of course, if one wanted to cheat to automate them... :) but it's easy to get addicted and become a stereotypical neckbeard living in the basement, eating hot pockets and mountain dew, playing wow or lol or whatever all day fucking long instead of trying to get laid or doing schoolwork. Way too easy.
Spending money? Eh. Depends on what you can afford. $60/month is certainly more than I got but is not the most outrageous sum I've heard. Compromise and pay your kid for doing chores that would otherwise be far more expensive. Cleaning the entire house, mowing the lawn and landscaping and taking care of scrap, putting out the garbage and bringing it in, cleaning the garage, washing and waxing the car, cooking... fixing those little things that need fixing but aren't dangerous (rattling knobs, paint touch-ups, sanding something down, replacing bulbs, whatever.) Pay for that, if you can afford it. Sounds like a lot of work but for, say, a hunnert a month, I would have been all over that as a kid. Cheaper than replacing things with small defects. Cheaper than hiring someone to do pretty much anything. Plus, as they grow older, they can do harder things; I remember replacing all the electrical sockets in my parents' house - an electrician quoted something like $75/per, but it's super easy to do yourself. I'm actually sympathetic to this, even though I got a lot less as a kid.
Sports etc? If you can afford it, sure. And as long as it's for the right reasons. The wrong reason is "Because I always wanted to do x" or "it looks good on a college application." For fun, health, strength, passion, discipline, and becoming a better person, go for whatever can be afforded.