Why do languages have irregular verbs?

Because the irregular ones are the oldest and most frequently used, and frequency protects them from the regularising pressure that flattens everything else. Irregularity is not untidiness — it is a fossil record.

The mechanism. Regular patterns spread by analogy: speakers encountering an unfamiliar verb apply the default rule. A rare verb gets regularised because nobody remembers its odd form, so they reach for the pattern. A very common verb is heard thousands of times, its irregular form is learnt directly, and it resists.

The result is strikingly predictable. In English, the irregular verbs are be, have, go, do, say, make, take, come, see, get — precisely the everyday ones. Studies of English over a thousand years have found that irregular verbs regularise at a rate related to their frequency, with rarer ones changing far faster. Holp became helped; chid became chided; be has not budged in millennia, and remains irregular in most related languages.

Where the strangeness originally came from. Many English irregulars are remnants of an older system called ablaut, in which vowels changed systematically to mark tense — sing, sang, sung; drink, drank, drunk. That system was once regular and productive. Time eroded it into scattered survivors.

Other sources of irregularity:

Suppletion, where a single verb's forms come from entirely different original words. Go and went are unrelated — went was the past tense of wend. Good and better likewise. This is common across languages for exactly the most-used words.

Sound changes applying unevenly, leaving related forms looking unrelated.

Borrowing from languages with different patterns.

Why children overregularise. A child saying goed or bringed is demonstrating that they have extracted the rule, which is a developmental advance rather than an error, and they later learn the exceptions individually.

Why new verbs are always regular. Google, text, download all take -ed. The productive rule is the only one still operating.