How to explain molecular geometry without the help of VSEPR, valence bond, or hybridization theories?
Short Answer: No, you can't. Reason being that so far this is the only way devised, which works most of the time.

Long Answer:
The Molecular Orbital Theory is, in practice, a complement to the Valence Bond Theory, Hybridization and VSEPR. 

There are successes as well as shortcomings of both the Molecular Orbital Theory, and the other three.

In general, the VSEPR theory is used to predict shapes of molecules and ions, not the Molecular Orbital Theory. It can be extended to a large number of Covalent compounds, but it has a few limitations as far as ionic compounds and multi-centered bonds are concerned. The MOT tells us that when atoms combine, they form molecules orbitals, which are lower in energy than the atomic orbitals. It is generalizable, and can be extended to even semiconductors and superconductors. It is very useful when it comes to finding the actual energies associated with every single orbital.

The Valence Bond Theory, along with Hybridization and VSEPR, is the one that is generally used to find the shapes of molecules. The reason is simple: The Molecular Orbital Theory tells us the arrangements of delocalized electrons in Molecular Orbitals, whereas, the other theories tell us the possible positions of the combined atoms after considering that the electrons are localized and repel each other.

However, there is the case of multi-centered $\pi$-bonds, also known as banana bonds, and the 4c-3e bonds, which can be explained only by the Molecular Orbital Theory. A famous example would be Di-Borane, or $\ce{B2H6}$.

Thus, both the Valence Bond approach as well as the Molecular Orbital approach are often required for an accurate prediction of the shape of a molecules.

I quote an article from Chemistry LibreTexts:


  Both the MO and VB theories are used to help determine the structure of a molecule. Unlike the VB theory, which is largely based off of valence electrons, the MO theory describes structure more in depth by taking into consideration, for example, the overlap and energies of the bonding and anti-bonding electrons residing in a particular molecular orbital. While MO theory is more involved and difficult, it results in a more complete picture of the structure of a chosen molecule. Despite various shortcomings, complete disregard of one theory and not the other would hinder our ability to describe the bonding in molecules.


Thus, although the VB, VSEPR, and Hybridization theories are outdated and have many limitations, they, along with the Molecular Orbital Theory provide us with a toolbox for almost any molecule possible.

(If there is any place where I have erred, please correct me)