What are the reasons why chromium and copper have half-filled #4s# orbitals?

A completely filled set of valence orbitals occupies all the quantum states for that sublevel. That is stable with respect to incoming atoms that would have donated or accepted electrons.

There is nothing inherently special about having a half-filled subshell. That is not something sought after by atoms, and if any textbook describes it that way, they are omitting some truth.

CHROMIUM

The most acceptable reasons why chromium acquires a #4s^1 3d^5# valence configuration are elaborated on here and here.

As it turns out, without relying on half-filled subshell business, the last orbital for a valence electron to go into #"Cr"# is the #4s#, and the resultant energy of the #4s# electron is lower than those in the half-filled #3d# orbitals. The half-filled subshell argument clearly fails for #"W"#, whose valence configuration is #6s^2 5d^4#.

COPPER

Copper is more simple. Consider how the #3d# orbital energies change across the periodic table (Appendix B.9).

The #3d# orbital energies for #"Cu"#, specifically, are significantly lower than the #4s# orbital energies, so there is no good reason for copper to have had a configuration of #3d^9 4s^2# instead of #3d^10 4s^1#, especially because the added coulombic repulsion would have been generated in a significantly higher-energy orbital.