Interesting choice using the tragedy of the commons metaphor and approaching it from your perspective. The property system in the US often looks to the English enclosure movement and their privatization "fix" of the overuse/underdevelopment problem inherent with the commons. However, it is also cast as a response to the failure of older educational/communal behavior control from scaling with growing populations. The view was that agrarian "land ethic" type self-motivated behavior control might've worked in smaller communities, but they couldn't keep up. The commons used to work, but it became a tragedy when the population outgrew the system. Hence, larger legal institutions like privatization and regulation stepped in.
Privatizing split up the commons, changing the incentive structure to encourage stewardship of each individual's portion. The cost/benefit calculus of your herdsmen changes if they both own the sheep and the grassland. There's more to it, but that's the gist.
If privatization misses something, regulation ideally fills in the gap. Ironically, your image of two herdsmen racing to expand their herd reminds me of a hypothetical involving oil wells. Oil, weirdly enough, is managed by similar property laws as wild game: if it "comes onto your land" and you're the first to "catch it" then it's yours. In practice, this means you can place a drill on the surface of your land and any oil it sucks up is yours - including oil that might have previously sat below your neighbor and "drained" onto your land (just google Daniel Day-Lewis Milkshake). This dynamic sets up a bit of an arms race. Each neighbor adds far more wells than they would have otherwise and the oil drilling becomes economically and physically inefficient. So, where mere privatization fails, government regulation steps in and says: oil wells need to be spaced 'x' distance apart and setback from the property line. There are other, more creative ways the government might regulate that market, but you get the point.
I'm not sure if privatization extends well to climate change. Parsing up the global ecosystem into alienable private property units feels like a very bad idea. However, I don't think we should sleep on the importance of regulation. Education is important. I stan Aldo Leopold as much as the next guy. Still, we've hit a point where most people's Land Ethic has more to do with the policies/regulations they vote for than running your farm or restoring an orchard with your family. Maybe we can establish a community of environmental superstars. That'd be great. For everyone else, we're going to have to use carrots and sticks.
