Why are irregular “to be”s quite different?

The reasons for the striking irregularities of the copula verb in Indo-European are addressed in Did the present indicative forms of the Latin verb “esse” evolve from two different roots? As those answers show, the Proto-Indo-European verb was in fact almost regular.

But there is a broader question of why the copula in particular is persistently irregular cross-linguistically. That has to do with the fact that it is the most frequent and the least contentful verb in a language.

Because it is underspecified, it attracts more suppletion, such as can be seen in the merger of be and was in English. Because it is so frequent, it is subject to much more phonological pressure towards ease of pronunciation and erosion. And because it has such a core function, language learners use it as a fixed set of given words, seared into their brains, and they resist any effort within the system to regularise it. That is why irregularities persist only in very frequent nouns and verbs.