I think it's just called the "broken stick" rule. Specifically, if you break your stick, you HAVE to drop it on the ice immediately and be done with it. If you even skate back to the bench holding it, you can get called for a penalty.
I have seen maybe a total of 15 minutes of hokey in my life so sorry if the question is dumb, but I would expect that there is a risk to stumble upon the stick?
In volleyball, if a ball rolls within the court, the play is stopped immediately (this happens quite often when you play on several courts side by side)
'Substitutions' being on the fly too (and <1m apart), there's just something kind of raw and basic about it. Obviously there's loads of rules, but the general gist/feel of it just seems a lot more.. I don't know, 'informal', or something, than other games.