The intuition behind carbon's covalent bonds

As an introduction to hybridization, my textbook discusses that "The formation of covalent bonds often starts with the excitation of the atoms." Taking carbon as an example, the author says:

Carbon forms four covalent bonds. Yet if we consider the elctron configuration in the carbon atom, we would not predict this as it has only two singly occupied orbitals available for bonding. ($1s^2\, 2s^2\, 2p_x^1\, 2p_y^1$)

I don't understand why carbon's bonding is counterintuitive. Why don't the 4 added electrons just go into the $p_x$ and $p_y$ orbitals then into some newly formed $p_z$ orbital? Why does the existence of "only two singly occupied orbitals available for bonding" complicate things?

As an extension, why is the excitation of the s-orbital electron to $p_z$ needed?

Thank you for your answers and please try to respond keeping in mind that I am a high schooler (more specifically, part of the IB Diploma program).

The most naive answer would guess that since every orbital wants two electrons and there are two orbitals with only 1 electron in each, then 2 more electrons are needed.

However carbon typically combines the one $2s$ orbital and three $2p$ orbitals to form four equivalent $2sp^3$ orbitals each with one electron. The four hybrid orbitals have a tetrahedral shape. So after hybridization, the carbon atom wants to create four bonds to fill the four electron shell vacancies.

My answer is very similar to MaxW's answer, but maybe it can still provide a bit of insight.

Consider methane as our current molecule: 1 carbon atom reacts with 4 hydrogen atoms to form 1 methane molecule. Most/Many textbooks explain this by hybridization of three p orbitals and one s orbital to form four energetically equivalent (degenerate) sp3 orbitals. This is the valence bond model.

"[Carbon] has only two singly occupied orbitals available for bonding."

The author suggests that since only two electrons are available carbon should form two bonds. This would leave the last carbon p orbital completely empty, though, and, tbh, this would to me be completely counterintuitive. To say it in a sloppy way: Chemical bonds/orbitals are all about being either completly full or completly empty (half-filled is also possible sometimes). Leaving one of the three p orbitals completly empty is absolutely not what any chemist would expect in such a situation. To suggest that carbon would only form two bonds is actually the really counterintuitive thing here (at least for me). It would even make more sense that carbon loses its two p electrons in order to have noble gas configuration, but ok... let's leave that aside

"Why don't the 4 added electrons just go into the px and py and some new pz?" The thing is that every bonding orbital can contain maximum two electrons. A bond between two atoms is only established, if every atom provides one electron each. If one carbon p orbital is empty then it would not make sense to put in two electrons from hydrogen. If two hydrogen electrons are put in the same bonding orbital it only means that the two hydrogen atoms are now bonded and form an H2 molecule.

So since every electron from the 4 hydrogen atoms needs an electron from carbon, the best way is to take the two electrons from the 2s shell. In that way carbon can provide its two 2s electrons for two bonds with two hydrogen atoms and the 2 p electrons can bind with another two hydrogen atoms. This will give four bonding orbitals, which are all completely full.