MSBuild not restoring NuGet packages - msbuild

Question

I have a solution consisting of multiple projects, several of which reference NuGet packages. There's a web site but also a service which has to be built for x86 or x64; the service is what I'm currently trying to build.
We're on TFS 2013; for this particular solution, I'm using Visual Studio 2015.
"Allow NuGet to download missing packages" and "Automatically check for missing packages during Visual Studio build" are both checked.
I was running NuGet 3.4.4, I've updated to 4.6.2.
I can build fine from Visual Studio. But automated builds on the server fail; but more importantly, it also fails if I try to run MSBuild locally. Similar to this question and this one.
As far as I can tell, the release template is the correct version (12), but I don't think it's a server issue, since I can't build locally either.
The packages are not in source control.
I have a nuget.config in the solution root folder; it previously was in a .nuget folder under the solution root. I do not (and never did) have nuget.exe or nuget.targets in the nuget folder. Here's the config:
<?xml version="1.0" encoding="utf-8"?>
<configuration>
<packageSources>
<clear />
<add key="NuGetCache" value="\\BuildServer\NuGetPackages" />
<add key="NuGetV2" value="https://www.nuget.org/api/v2/" />
</packageSources>
<packageRestore>
<add key="enabled" value="True" />
<add key="automatic" value="True" />
</packageRestore>
<solution>
<add key="disableSourceControlIntegration" value="true"/>
</solution>
</configuration>
If I (manually) clear the packages and run
MSBuild /p:Configuration=Release;Platform=x64
(from a VS2015 command prompt), it fails, because it can't find the referenced packages.
As suggested in this answer, I've removed the EnsureNuGetPackageBuildImports targets from my projects; they did not have a RestorePackages tag or an import for nuget.targets.
I tried adding a pre-build event to do the package restore (yes, I realize this is the "old" way of doing things)
"C:\Program Files (x86)\NuGet\nuget.exe" restore -ConfigFile "$(SolutionDir)nuget.config" -PackagesDirectory "$(SolutionDir)packages"
... but it consistently fails ("The command exited with code 1"). If I copy the NuGet command line from the MSBuild output and execute it on its own, it succeeds.
I have noticed one odd thing: in the pre-build event, NuGet says "The folder d:\path\to\solution\project1\bin\Release-x64 does not contain an msbuild solution or packages.config file to restore". I'm not sure why it's looking for those in the output folder.
I do have a workaround for this. We have a network share set up to use as sort of a package cache. If I hand-edit the project files and change each reference so that the HintPath points to \\BuildServer\NuGetPackages\ rather than ..\packages\, the build will succeed. But this is ugly, and more importantly it significantly degrades the performance of Visual Studio.