Found a solution! (Thank you, Esri support.) In Catalog, go to the mosaic dataset, right mouse for properties, Defaults tab, change Max # of Rasters per Mosaic to be larger than the number of your source images (the default is 20). And change Max # of Records Returned per Request to a ludicrously large number, like 100000. They didn't know why it helps... but it did.
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