Why in MuSiC (DoA) don't we search for minima of the inverse (pseudo)spectrum?
Let's define the null-spectrum $Q(\theta)$ and pseudo-spectrum $P(\theta)$ as
$$Q(\theta) = a(\theta)^H{v_n}{v_n^H}a(\theta)$$
$$P(\theta) = \frac{1}{Q(\theta)}$$
The choice to find the maxima in $P(\theta)$ comes down to personal preference. It is more intuitive to associate a peak in a signal to some phenomenon being observed. This is especially so when reading a text, where the author is trying to teach you how the algorithm works.
You can very well search for the minima in $Q(\theta)$ instead of the maxima in $P(\theta)$ and get the same answer as can be seen in the example below


Nothing is stopping you from using the minima. In general there is no significant advantage to using either except for avoiding division operations and potential divide-by-zero, which usually fine in real systems where some kind of noise is present. Still, certain systems and their implementation can make using one over the other more appropriate.