If there is no current limiting, the diode will heat until it would burn, and you'll have a short circuit or open circuit, and it won't work.

What actually prevents me from using a regular diode as a zener one, given that I'll take care not to burn it with excessive heat?

The breakdown area will not damage a diode, if the current is limited. Zener diodes constantly work in this mode.

There are many differences. First, Regular diodes generally have much higher breakdown voltages than zener diodes. Next, Regular diodes tend to experience avalanch breakdown.

Thanks for your intelligence. Pretty interesting and useful.