Why do phosphorus and sulfur have variable valences?
For a modern answer (with some hand-waving), we can simply say that valence electrons occupy more space as we go from first period to second period to third period. So hydrogen makes one bond, second period elements up to four, and third period elements up to six.
If you want to stick to Lewis structures and hybridization, you can explain $\ce{PCl5}$ with resonance:

Source: https://www.cusd80.com/cms/lib6/AZ01001175/Centricity/Domain/1108/003-center%204-electron%20bond.pdf
If you like molecular orbital diagrams, it is pretty straightforward to expand a $\ce{EX3}$ MO diagram to a $\ce{EX5}$ diagram without using d-orbitals, e.g. http://www.chembio.uoguelph.ca/educmat/chm364_preuss/5_5contd2%20MO%20Theory_ICl4%20PCl5.pdf. Like in the valence bond model, the bond order will decrease as you increase bound atoms.