I seem to have solved it, though I'm still not sure why the code in my original question does not work.
Instead of specifying paths via concatenation (e.g. AssemblyFiles="%(PackageDir.FullPath)\MyAssembly.dll") I put each path into its own item:
<ItemGroup>
...
<PackageVersionAssembly Include=".\build-artifacts\package\MyAssembly.dll"/>
<NuSpecFile Include=".\build-artifacts\package\MyAssembly.nuspec"/>
...
</ItemGroup>
I made the same change in the task and made the same change to references to the .nuspec file.
The new Package target looks like this:
<Target Name="Package">
<GetAssemblyIdentity AssemblyFiles="@(PackageVersionAssembly)">
<Output TaskParameter="Assemblies" ItemName="AssemblyIdentity" />
</GetAssemblyIdentity>
<XmlUpdate
Prefix="nu"
Namespace="http://schemas.microsoft.com/packaging/2011/08/nuspec.xsd"
XmlFileName="@(NuSpecFile)"
XPath="/nu:package/nu:metadata/nu:version"
Value="%(AssemblyIdentity.Version)" />
</Target>
I hope this helps others!