Is it normal for drawer boxes to have a big chunk cut out of the front corner?

That notch is a real thing and it's usally done on purpose. When a drawer sits above or beside a door on the same cabinet, the hinge arm swings back inside the box when the door opens, and if the drawer box is full depth it'll smack right into it. Shops notch the corner to give the arm room to travel. The fact that it's a clean rectangular cut that lines up exactly with the hinge tells you it was planned at the shop, not hacked in on site. And since your drawer still runs smooth with no wobble at the cut, structurally you're fine. Most drawer boxes lose very little strength from a corner notch like that.

I had something similar in my old kitchen and thought the installer broke the drawer, untill I opened the door next to it and watched the little bendy arm swing right into that empty space. Made me feel less crazy that this thread exists.

One thing I did after mine was stick my phone in the drawer with the flash on and watch the door close from inside, just to see exactly where everything moves. Helped me stop worrying about it. Does the cut corner show at all when the drawer is pulled out, or is it hidden behind the bin inside?

Yeah this is standard, and honestly if anything you got the better version. I did a full kitchen reno on my sister's place two years ago and her cabinet shop skipped the notch entirely, just spec'd shallower drawers on the doors side. Fine until she wanted to store bigger stuff in those and suddenly the 'why are my drawers short' conversation starts. A clean cut corner is invisible once the bin's in there. One thing to check though, peek at wether the notch got any edge banding or sealer on the exposed edge. On mine they left one raw and it swelled up slightly after a couple of humid summers. A quick wipe of finish fixes that if it's bare.

Same thing on my build last spring, except the shop skipped the notch and the drawer was alredy installed when I found out. The door chewed a chunk out of the drawer side instead, which is how I learned that lesson the hard way. Only thing I'd check on yours is the raw edge of the cut, mine came edge-banded from the factory so it never swelled or fuzzed up. If yours is bare plywood I'd hit it with a dab of clear finish just so it doesn't get rough from damp dish towels sitting against it.