I'd treat this as a Bayesian inverse probability problem. Per Laplace's rule-of-succession solution of the sunrise problem (see https://en.wikipedia.org/wiki/Rule_of_succession and https://en.wikipedia.org/wiki/Sunrise_problem), it follows that if an engine wins n out of n games, its rating should be estimated as if it played n+3 games, winning n+1, drawing one, and losing one - playing, of course, against the same pool of opponents. (A point of note: a chess game differs from a sunrise in that there are three possible outcomes, rather than just two. For the sake of simplicity, we'll assume the number of games with black and white pieces is equal.)
Using the example provided above, the rule-of-succession method returns an estimate of 2230 (1000 games), 3330 (1M games), and 3450 (2M games) - not that far off from Bayeselo. I'm not familiar with the method employed by Bayeselo and cannot say how mathematically sound it is, but in the end this is a kind of problem that doesn't have a "correct" solution because it hinges on one's choice of prior probability. In Laplace's approach, it is posited that, if we knew nothing about the history of sunrises, we should assume that the probability of Sun rising tomorrow is uniformly distributed between 0 and 1. Whether that assumption is applicable to a chess game between two players about whom we know nothing is anyone's guess.
(Additional note: I'm assuming here that the question is how to estimate Elo once n games have already been played, rather than when to stop playing more games. The answer to the latter question is significantly more complex as it obviously depends on additional factors.)