I must admit it doesn’t work the way I thought it did, either. based on the Block description, I thought one drag-and-dropped related records directly in the Page Designer edit window to change their order. Instead, from the Page Designer window, one selects ‘Expand record’ to open the traditional expanded record screen and reorders the sequence of linked records there. (Presumably this means the sequence has to be set manually for each record in the base; unfortunately, the table in my ‘scratch’ base I used to check this has only a single record with a many-to-one link.)
And, yes, you are right: Airtable treats related records as a LIFO (last in, first out) stack, so items are returned in reverse order from how they were added. This means, with sufficient omniscience, one can simply add related records bassackwardly have them presented in proper order.