Действительно ли возможно установить частную собственность через отражение?

Да, это так:

/// <summary>
/// Returns a _private_ Property Value from a given Object. Uses Reflection.
/// Throws a ArgumentOutOfRangeException if the Property is not found.
/// </summary>
/// <typeparam name="T">Type of the Property</typeparam>
/// <param name="obj">Object from where the Property Value is returned</param>
/// <param name="propName">Propertyname as string.</param>
/// <returns>PropertyValue</returns>
public static T GetPrivatePropertyValue<T>(this object obj, string propName)
{
    if (obj == null) throw new ArgumentNullException("obj");
    PropertyInfo pi = obj.GetType().GetProperty(propName, BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance);
    if (pi == null) throw new ArgumentOutOfRangeException("propName", string.Format("Property {0} was not found in Type {1}", propName, obj.GetType().FullName));
    return (T)pi.GetValue(obj, null);
}

/// <summary>
/// Returns a private Property Value from a given Object. Uses Reflection.
/// Throws a ArgumentOutOfRangeException if the Property is not found.
/// </summary>
/// <typeparam name="T">Type of the Property</typeparam>
/// <param name="obj">Object from where the Property Value is returned</param>
/// <param name="propName">Propertyname as string.</param>
/// <returns>PropertyValue</returns>
public static T GetPrivateFieldValue<T>(this object obj, string propName)
{
    if (obj == null) throw new ArgumentNullException("obj");
    Type t = obj.GetType();
    FieldInfo fi = null;
    while (fi == null && t != null)
    {
        fi = t.GetField(propName, BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance);
        t = t.BaseType;
    }
    if (fi == null) throw new ArgumentOutOfRangeException("propName", string.Format("Field {0} was not found in Type {1}", propName, obj.GetType().FullName));
    return (T)fi.GetValue(obj);
}

/// <summary>
/// Sets a _private_ Property Value from a given Object. Uses Reflection.
/// Throws a ArgumentOutOfRangeException if the Property is not found.
/// </summary>
/// <typeparam name="T">Type of the Property</typeparam>
/// <param name="obj">Object from where the Property Value is set</param>
/// <param name="propName">Propertyname as string.</param>
/// <param name="val">Value to set.</param>
/// <returns>PropertyValue</returns>
public static void SetPrivatePropertyValue<T>(this object obj, string propName, T val)
{
    Type t = obj.GetType();
    if (t.GetProperty(propName, BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance) == null)
        throw new ArgumentOutOfRangeException("propName", string.Format("Property {0} was not found in Type {1}", propName, obj.GetType().FullName));
    t.InvokeMember(propName, BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.SetProperty | BindingFlags.Instance, null, obj, new object[] { val });
}

/// <summary>
/// Set a private Property Value on a given Object. Uses Reflection.
/// </summary>
/// <typeparam name="T">Type of the Property</typeparam>
/// <param name="obj">Object from where the Property Value is returned</param>
/// <param name="propName">Propertyname as string.</param>
/// <param name="val">the value to set</param>
/// <exception cref="ArgumentOutOfRangeException">if the Property is not found</exception>
public static void SetPrivateFieldValue<T>(this object obj, string propName, T val)
{
    if (obj == null) throw new ArgumentNullException("obj");
    Type t = obj.GetType();
    FieldInfo fi = null;
    while (fi == null && t != null)
    {
        fi = t.GetField(propName, BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Instance);
        t = t.BaseType;
    }
    if (fi == null) throw new ArgumentOutOfRangeException("propName", string.Format("Field {0} was not found in Type {1}", propName, obj.GetType().FullName));
    fi.SetValue(obj, val);
}