Yes - I thought you were trying to restate the gluing argument - my mistake - I also think that the bundle argument which is very neat does not have an independent proof - you need to derive it from the other arguments - if I am wrong about this I would love to see the direct bundle argument - it would be very cool -I thought you had already figured it out - I was trying to rephrase it terms you might be more familiar with. The natural projection from the fundamental group gives you an index 2 subgroup in the fundamental group, which a fortiori is normal. In this case, modding out the universal cover by this subgroup gives you a two-sheeted covering space of M. (cf. Hatcher's algebraic topology book, pp. 70-80).