In the literature one finds (at least) two approaches towards proving that the sample mean is uniformly minimum variance (UMV), among unbiased estimates that "ignore the labels," for the finite population mean: The "traditional approach" and the "scale-load approach." The identity of results under the two approaches extends to a more general setting, as shown in this paper: The Horvitz-Thompson estimate is UMV unbiased for any given fixed effective size design.
"On Uniformly Minimum Variance Estimation in Finite Populations." Ann. Statist. 4 (5) 993 - 997, September, 1976. https://doi.org/10.1214/aos/1176343598