It's not just the metaphor that's weak, it's also the underlying dichotomy
I have never heard the ideas in this metaphor expressed in this way before. I think there are better metaphors, but in order to find and understand them it's better to abandon the phrasing of "involvement vs. commitment." A light reading makes the whole thing sound like you should turn your office into some sort of crazy suicide cult to get the breakfast made. That's actually the opposite of what it's supposed to convey, that the "committed" should exercise more say over how things get done than the "involved." It sounds like that's really what your manager wants people to feel, but being compared to livestock probably paints the wrong picture.
Instead of a dichotomy, just focus on the thing you really want: Empowerment
It ultimately does not matter if some people are merely "involved" while others are "committed", you need everyone to start acting like thinking adults who can make decisions. In order to do that, they have to feel like they're allowed to make decisions. The common management buzzword for this is empowerment, that they feel empowered to decide things.
The best way to do this, is to take an attitude that lets employees know that they actually get to make the decisions, by making them make decisions. "What tools are we allowed to use for QA?" "You tell me. If you need us to buy it, let me know and I'll get some budget allocated." They'll understand it once they realize that, they're allowed to decide and that deciding leads to things actually happening. It's hard to explain it to people, it's something you learn through action.
If you still need metaphors because you feel the need to explain this to people, use ones that put the burden of deciding or acting on the employee
Here's a bunch:
- "You're the one in the driver's seat"
- "You guys are the boots on the ground here"
- "It's going to be your scalpel, not mine" -- specifically for something tool-related.
In all of these, the burden for acting is shifted to the employee; driver must steer, the soldier has to attack the enemy, the surgeon has to do the cutting. The problem with the livestock analogy is that livestock doesn't do anything besides stand around until it's time to get butchered.