So, Why can't three atoms share an electron and overlap their orbitals?
The normal case for a covalent bond is indeed a 2-centre 2-electron bond.
There are however cases, where 3 centres (= atoms) either share 2 or 4 electrons.
- In organic chemistry, the most famous case for an electron-deficient 3-centre 2-electron bond is the 2-norbornyl cation.
- In inorganic chemistry, boranes with their banana bonds are the role models.