The "short-term" effect on warming is quite severe and very relevant. Also, while carbon atoms are conserved, methane is not. You can make a lot of methane from biomass without needing to absorb methane to produce the biomass.
Accelerating the "natural" cycle is bad even if there is no net increase. A massive forest fire technically doesn't cause any net increase in CO2, but releasing that much CO2 at once causes a lot of problems that wouldn't happen if the same amount were released over hundreds of years as those trees died and decayed.
I'm sure some people would argue that burning oil and gas isn't a net increase either, because all of that carbon came from the atmosphere originally. The important question is what emissions will make life worse for humanity, and what can be done to avoid them. Avoidable methane emissions cause real-world problems regardless of their source.
> the question is still whether that carbon came from a fossil source or was part of the cycle to begin with.
I don't think that matters anymore. At some point, we wanted to reduce atmospheric CO2 to a tolerable level and an obvious approach was to not add to natural emissions.
Now we just need to reduce CO2 in the atmosphere. It doesn't do different things based on its source. If we can reduce it by cutting gas production or by cutting some natural source, we should do it.
In your thought experiment the number of extant cows is a variable determined by human behavior as opposed to inherent regulation. There might be historical population numbers from specific regions where carbon was in equilibrium but that's certainly not where we're at today nor is it a goal of ours.
I think we're only able see some carbon sources as "outside the cycle" due to our infinitesimal lifespans.
Let's say a cow eats grass, produces methane, the methane gets converted into CO2, the grass grows back. No net increase in CO2.
If, on the other hand, the methane came from natural gas, you have permanently added carbon to the cycle.