If the sampling process gives you a random subset of tiles, I don't see how the estimate could be biased. Even if you have a strange distribution of circles (heavily clustered in certain geographies), the weak law of large numbers would mean that the sample mean is an unbiased estimator for population mean if your sampling process is iid (random number generation is by definition iid).
Ugh, you are of course correct, I should not be allowed to comment on statistics before having my first coffee. I think I had some diffuse ideas about this not being a poisson processes in my head that lead me to this faulty conclusion.
https://en.wikipedia.org/wiki/Law_of_large_numbers
I think you're confused between which restrictions apply to the sampling process versus the underlying population.