>you'll quickly realize that it changes on a predictable ~365-day loop.

Quickly, being relative here.

In the beginning of my mortgage, first 2-3 years, they couldn't properly predict my property taxes (sometimes predicting $0 in a quarter) and had wild swings in escrow amount. Started with a $6000 refund year 1, then lowered the monthly amount, then went high with a $400/month increase.

Next year it should go back down a bit and settle out for the future.