Why can a wave-function be written as a product of 1-electron wave-functions?
To start with your first sentence: "Why can a wavefunction be written as a product of 1-electron wave-functions"? Well, you can write whatever you want to approximate a wavefunction; physics police is not going to come and take you away. But the two important questions are: How useful will the results from your approach be, and how hard is it to work with mathematically? These criteria obviously reduce your number of options quite a bit.

A wavefunction that is constructed as a product of one-electron orbitals definitely has that second point going for it. As you have already experienced yourself, it is straightforward to write down and calculate with. Arguably the most important implementation of this approach is Hartree-Fock theory where the electrons never see each other as distinct point charges in space, only as fuzzy charge distributions that each one has to wade through individually. Hartree-Fock gives you fairly straightforward equations to work out the optimal way to build the independent-particle wavefunction; and while it cannot accurately capture the energetics of interactions among the electrons "out of the box", it's a clean and simple starting point to describe the quantum mechanics of your molecule. And it's often not all too bad, either - for typical well-behaved cases, Hartree-Fock and its one-electron orbitals can already land you within a few percent of the full energy value. The remainder is still way too large for it to be practically useful, but you can reuse this initial wavefunction in more refined algorithms to reduce the error further.