Found the solution this morning.

1. I converted my packages.config to package reference See docs
2. I updated my csproj of the NuGet package to be:

<None Include="Xsds\*.xsd">
  <pack>true</pack>
  <PackagePath>contentFiles\any\any\Xsds\</PackagePath>
  <PackageCopyToOutput>true</PackageCopyToOutput>
  <CopyToOutputDirectory>Always</CopyToOutputDirectory>
</None>

I would like to know if this is possible to do with a packages.config. It's rather annoying that this NuGet package simply won't work unless the user is doing it a specific way..