Update (2/26/17): It turns out that I did have uses for several of the GDAL 2.1 features, and there was a simple fix for the problem — switch to a different distribution. I tried the GIS Internals msi installer and everything worked the way it should.
I was preparing to load a US county shapefile to a SQL Server table today and I needed a copy of ogr2ogr in order to create the shape table and convert the SRID. I did the obvious thing, which was to download and install the latest version (2.x) of GDAL and, lo and behold, it didn’t work, failing with some really obscure error. Several Googles later I realized that I was going to have to fall back to version 1.x, so I installed that and, lo and behold, it worked just fine.
I’m sure glad that I am not trying to do anything fancy that would require the latest version of GDAL or ogr2ogr.
My SQL Server version is a recent CU of SQL Server 2016 (Dev Essentials download).