But, what about indoor? Low lit conditions? Gear absolutely does matter!

Not really, no.

Your 600D is perfectly capable of shooting decently in low light. Cameras older and less advanced than yours have been (and still are) used to capture images in far more difficult and challenging conditions than a child's birthday party.

I do have a 50mm f/1.4, but I couldn't use it much because it was a birthday and there wasn't much time for me to keep switching lenses. It was also a small venue, so the 50mm was a very restricting focal length, forcing me to go wide.

You're just making excuses. People who are committed find ways to change lenses. (It's really not that difficult.)

That 600D is a fine camera. I'm sorry to say, but the problem isn't with your camera. Your frustration comes from the fact that you spent money on a camera and since you don't actually know how to use it, you're not getting good pictures. So you throw up your hands and say "I NEED BETTER GEAR" when the reality is, you need better skills.