Why are equations of state for a non-ideal gas so elusive?
It's been known since 1941 that the answer to your question is in the negative, i.e. that there will never be a closed form equation of state for a nonideal gas.

In 1941 Mayer and Montroll developed what is now known as the cluster expansion for the partition function of a nonideal gas whose particles have pairwise interactions. This cluster expansion provides a 1-1 correspondence between various integrals over the interaction potential and virial coefficients in the Kamerlingh Ohnes equation of state which has in principle an infinite number of terms in it. Therefore, it would not be considered a closed form equation of state. In practice the virial coefficients are known to decay so it is usually safe to truncate the expansion for practical calculations.

One could extend the cluster expansion to three-body and higher-order interaction potentials but this will not change the fundamental argument above.