The problem isn't the cycle, it's the delta of total emissions in the cycle. By your definition, literally everything is part of the carbon cycle, as we are just putting the carbon of old plants in the air, which will slowly be consumed by plants. The problem is if we put all that carbon in the air all at once, we have problems.
The biomass of livestock is 14x larger than all other mammal apart from humans[1], so it makes sense that, even if their carbon cycles is short, it's still a massive amount of effectively permanent GHG that exists in our atmosphere that wouldn't otherwise be there... about 15% of all emissions[2].
Not counting livestock as emissions because they form a decades long closed loop could be fine when we are carbon negative, but we are dealing with the very real problem of total emissions right now, not just unsustainable growth of emissions.
The biomass of livestock is 14x larger than all other mammal apart from humans[1], so it makes sense that, even if their carbon cycles is short, it's still a massive amount of effectively permanent GHG that exists in our atmosphere that wouldn't otherwise be there... about 15% of all emissions[2].
Not counting livestock as emissions because they form a decades long closed loop could be fine when we are carbon negative, but we are dealing with the very real problem of total emissions right now, not just unsustainable growth of emissions.
1: https://www.pnas.org/doi/10.1073/pnas.1711842115
2: https://www.ucdavis.edu/food/news/making-cattle-more-sustain...