Isn't every photodiode an avalanche photodiode?

Most every semiconductor junction has an avalanche mode wherein a high enough reverse voltage causes a cascade of electron-hole pairs to be generated by impact ionization. So why wouldn't an ordinary photodiode behave like it's much more expensive cousin, the Avalanche Photodiode? They're often made from the same material (InGaAs).
Ordinary photodiodes sell for penny's. Avalanche photodiodes sell for > $100.
So why can't I use an ordinary photodiode with a high voltage high impedance reverse bias to greatly increase the quantum efficiency of the photodiode?

I think it's like how every diode is a capable of breakdown like a zener diode, but zener diodes are designed specifically to operate in breakdown.

Yes, an avalanche diode is specificially designed and characterized to operate in avalanche mode. A normal diode may or may not operate properly in the avalanche mode, and may be erratic or noisy. You'd have to try a particular device to see how well it works.