It's a valuable observation and there are many potential tangents to take in response. I like to test theories by looking for exceptions, and for example this season the Nationals are on a pace to score 875 runs, yet (as of this morning) are below .500 because they give up runs at an even higher pace. However they're merely the exception that proves the rule because yes, a typical pitching staff and the defense behind them will make 5 runs stand up more often than not. The context of a given season matters, as you point out, so that this year and the past two years the average runs scored has been about 4.4 per game but as recently as 2023 it was 4.6 so then your 5 runs gets you less of an edge than right now - and 2019 was even more out of kilter at 4.8. Long ago, Bill James popularized an empirical observation that every 10 additional runs scored during a season (and conversely every 10 runs saved) works out to about 1 additional win - a ratio that at first blush sounds kind of ludicrous since ten key runs here or there could net you several wins not just one, but the distribution of runs across a season doesn't work that neatly. Those are some random thoughts, and I could go further and further down this rabbit hole, but my main reaction to your message is simply "yup."
Still, I want to steer this back to variability, and your 810-run rule of thumb makes for an entertaining thought experiment. Last year two teams scored 810, the Yankees and the Dodgers, and everyone envies their win totals. As you say, if you score 5 runs a game in today's environment, you set yourself up for a successful season. Nobody scores exactly the same number each game, but if you did and it was 5, you'd do well. However, if 810 for the season is the target, then there are other ways it could happen. Suppose you scored 10 in half your games but got shut out (even when you go into extra innings, let's say) in the other half; you'd still score 810, but at best you'd finish .500, and since most teams, even good ones, give up 11+ runs a handful of times in a season, you'd be odds-on to finish something like 78-84. Let's make the variability even wackier: suppose you score 15 runs in a third of your games and get shut out in two thirds. That still brings you to 810 runs, and assuming your pitching/defense never completely blows up, you are destined for a 54-108 debacle of a season.
That's a thought experiment and can never happen - but neither does a constant 5 run output across an entire season either. The idea in the above paragraph is to demonstrate that variability could be really, REALLY important - the greater the variability the greater the potential effect - but as far as I know it's not really studied at all. And as I said earlier, someone more schooled in industrial-grade statistical analysis than myself would need to be called on to even begin the process of defining what to look for. Because batters, pitchers, and managers intentionally do things for competitive reasons that skew the uniformity of the data available for us to work with - the manager chooses relievers differently when the score is 5-4 than when it's 8-1, as a trivial example, and pitchers pitch differently to Matt Wallner when the game is not close and/or there is no one on base, for another.