Serious question - have you ever killed a deer? I'm not flaming, it's a real question. After a few seasons of experience you'll answer your question yourself.
There is zero chance in my experience that you'd get off three good shots at a deer flying through the bush. If your plan is to rely on firepower rather than marksmanship, I'm telling you you're having a fever dream. It's just not going to happen that way.
At very close range (>15 yards) buck is devastatingly effective on deer, but it still requires the same care in shot placement that a rifle does. I view hunting with buck as if I was hunting with a bow - they require the same care in use, have about the same effective range and kill in the same way, ie solely through exsanguination. I've killed deer at living room distance with a blackpowder coach gun and home made 4 buck, but that was knowing I had a VERY short effective range AND after a few months of working up loads at the range. Would I have taken a rifle instead? Absolutely. Would I let the lack of a rifle stop me from hunting? Absolutely not. Know your tools and work within their - and your - limits.
I would not hesitate to hunt deer with buckshot, but I would also spend many hours on the range and many $ in ammo ensuring I absolutely 100% knew my effective range. I would also NEVER pick a shotgun if I could use a rifle. That decision is simple physics and physiology.
View buckshot hunting as bow hunting, and you won't get yourself into trouble.
The three round shotgun limit doesn't apply to all provinces - it's actually a federal migratory bird regulation, but Ontario simplified things there by applying it to all shotgun hunting. There are several provinces where you can load whatever's legal per the Criminal Code. Despite being unfettered where I live now, I've still not found it's of any practical advantage. Crows and coyotes move awfully fast...