At least for GPKG (I have not tested any other format), they do not work with any remotely recent version of Manifold 9.
Puzzling, but I cannot replicate the above. Here is what I did:
1. I located an old machine not recently used, which had old GDAL and QGIS installed.
a. renamed GDAL intallation folder from OSGeo4W64 to -OSGeo4W64
b. renamed QGIS installation folder from ~\QGIS 2.18 to ~\-QGIS 2.18
2. Verified QGIS no longer launches.
3. Launch Manifold and verified GDAL no longer works.
4. Download from the Product Downloads page and put the sqlite-spatialite-x64 folder in the ~\bin64 folder.
5. Launch Manifold and verify gpkg works.
It's puzzling. I have another machine, a 32-bit notebook, which has never had either GDAL or Q installed. I'll try with that after lunch.
By the way, I tried with Q installed/not installed to eliminate the possible path that if the sqlite dlls from the downloads page don't work, it is having a Q installation that is installing alternate sqlite dlls in the PATH.
Just to be sure, I searched the disk drive for ALL instances of spatialite.dll and verified that they existed only within the ~\bin64 folder where I placed the latest download, or within the paths where I modified the names so any PATH variable references will not work (I also checked the PATH and other environment variables).
It is interesting to note that the OSGeo4W64 and also the QGIS 2.18 spatialite.dll files are date stamped 6/23/2016 1:27 AM, exactly the same as the date stamp on the spatialite.dll file provided by the product downloads zip.
The logical conclusion is that if somehow those .dlls are being used instead of the one downloaded, they are still the same thing.
It could be something else going on, a mirror image of the problem: consider the hypothesis that the dll being used is not the one you think, but a dll that comes earlier in the PATH.
Case A: In your statement of that hypothesis, you state the possibility that maybe the dll Manifold provides is wrong, but my test case was working because without realizing it I was launching a good dll installed by GDAL that came ahead in the PATH before the "bad" dll I got from the Manifold site.
Case B: But it could be the mirror issue... suppose the reason you had problems was because you had an old GDAL/QGIS/something else that placed a too-old spatialite.dll in the PATH before the dll downloaded from Manifold. In that case, instead of running the dll provided by Manifold you were running a too-old or otherwise problematic dll installed by something else. When you installed GDAL, it could be you placed a newer, OK spatialite.dll in the path ahead of the problematic dll and then suddenly it started working.
Like I say, I see no spatialite.dll files in the PATH ahead of the one I just downloaded from the Manifold site. I checked that by looking at environment variables and also by searching the disk for any occurrence of spatialite.dll. I might have missed something, but the above seems to indicate the spatialite files downloaded from Manifold are OK. I'm baffled why they work for me but not for you.
PS: Great photo today on the site. Some I love, some I don't but this one I like. I think the theme is crunching with many teeth at once lets you bite off a bigger chunk of data. :-)