So - Amazingly syncing those .csproj files actually worked (I did have to clear cookies for both admin and MVC sites first). But now the page tab loads without error.
Below is the part of the .csproj file that I had previously removed (when it referred to version 13.0.0) - I guess this is needed, even though all the .dlls are copied to a single folder during the build process.
As a FYI, the main MVC solution has 30 different projects, as it's based on the Helix architecture principles from Sitecore.
<Import Project="..\..\packages\Kentico.Xperience.AspNet.Mvc5.13.0.52\build\Kentico.Xperience.AspNet.Mvc5.targets" Condition="Exists('..\..\packages\Kentico.Xperience.AspNet.Mvc5.13.0.52\build\Kentico.Xperience.AspNet.Mvc5.targets')" />
<Target Name="EnsureNuGetPackageBuildImports" BeforeTargets="PrepareForBuild">
<PropertyGroup>
<ErrorText>This project references NuGet package(s) that are missing on this computer. Use NuGet Package Restore to download them. For more information, see http://go.microsoft.com/fwlink/?LinkID=322105. The missing file is {0}.</ErrorText>
</PropertyGroup>
<Error Condition="!Exists('..\..\packages\Kentico.Xperience.AspNet.Mvc5.13.0.52\build\Kentico.Xperience.AspNet.Mvc5.targets')" Text="$([System.String]::Format('$(ErrorText)', '..\..\packages\Kentico.Xperience.AspNet.Mvc5.13.0.52\build\Kentico.Xperience.AspNet.Mvc5.targets'))" />
</Target>
Thanks for your advice so far though David - It's appreciated.