With a headshot it would take a small 100# deer readily.
The .32 centerfires are perhaps my favorite cartridges for woods bummin and I'd like to have a levergun in such chambering. Great for small game up to coyotes, in a rifle format close range deer may be in the realm. But if your chosen cartridge, what ever it may be, is only powerful enough for consistent kills with head shots than you need more for general hunting. In case of emergency is one thing, shots of opportunity may be another but general hunting no way. And with tight seasons and controls for deer hunting in most jurisdictions, you either pack a deer gun if they're on the menu or you don't.
Now keep in mind, I'm not saying a .32-20 in a lever gun isn't enough for deer. Many have fallen to it but I have no personal experience. But if you only think it's good enough for small deer and only with head shots than you plain and simply don't use it for em.