The EMF is being converted to a bitmap by the Save operation.
An EMF can be successfully written thanks to this answer, with minor modifications.
using System.Runtime.InteropServices;
including in the class
    [DllImport("gdi32.dll")]
    internal static extern uint GetEnhMetaFileBits(IntPtr hemf,
        uint cbBuffer, byte[] lpbBuffer);
    [DllImport("gdi32.dll")]
    internal static extern bool DeleteEnhMetaFile(IntPtr hemf);
and modifying the code
private void button1_Click(object sender, EventArgs e)
{
    Metafile emf = null;
    using (var ms = new MemoryStream(Properties.Resources.blank))
    {
        emf = new Metafile(ms);
    }
    IntPtr h = emf.GetHenhmetafile();
    uint size = GetEnhMetaFileBits(h, 0, null);
    byte[] data = new byte[size];
    GetEnhMetaFileBits(h, size, data);
    using (FileStream w = File.
        Create("C:\\Users\\chrisd\\Documents\\emfbitmap1.emf"))
    {
        w.Write(data, 0, (int)size);
    }
    DeleteEnhMetaFile(h);
}
Of course, if it is not required to have the EMF in memory it can be written to disc directly.  E.g.
private void button1_Click(object sender, EventArgs e)
{
    File.WriteAllBytes("C:\\Users\\chrisd\\Documents\\emfbitmap1.emf",
        Properties.Resources.blank);
}