Learn English – “Why don’t you…” question with the verb “be”

As Peter Shor indicates in his comment, these are modal verbs and are found in other Germanic languages as well. Their unusual conjugation is due to their historical source:

Germanic modal verbs are preterite-present verbs, which means that their present tense has the form of a vocalic preterite. This is the source of the vowel alternation between singular and plural in German and Dutch. Because of their preterite origins, modal verbs also lack the suffix (-s in modern English, -t in German and Dutch) that would normally mark the third person singular form.

Preterite-present verbs are a very old group of verbs found in Germanic languages that in turn descended from a set of Proto-Indo-European verbs. Originally these were perfect-aspect verbs that referred to a present state.

The common characteristic is that the "present tense" of these verbs was conjugated like the "past tense" (essentially). In Modern English this now means that the present-tense "-s" ending is not used, and that modifiers like "to", "will", etc. can't be added. It happens that these verbs are all modals in Modern English, which probably helps preserve the unusual conjugation.

So this is a bit of a historical accident that made a lot more sense in PIE, and now is frozen that way.

This is what Sir Ernest Gowers has to say in "Fowler's Modern English Usage" (Oxford 1965) under the heading noun and verb accent:

When there is both noun and verb work to be done by a word, and the plan of forming a noun from the verb, or a verb from the noun, by adding a formative suffix (as in stealth from steal) is not followed, then one word may be called on to double the parts. In that case there is a strong tendency to differentiate by pronunciation, as in use (n. us, v. uz); such a distinction is sometimes, as in use, unrecorded in spelling, but sometimes recorded, as in calf and calve. Is is not possible to draw up a complete list of the words affected, because the impulse is still active, and the list would need constant additions, especially of words whose pronunciation can be modified without change of spelling.

He goes on to give partial lists of the two main forms that the question identifies; the largest class being words "whose accent is shifted from the first syllable in the noun to the last syllable in the verb"; the other group are words, especially monosyllables, which "are differentiated not by accent but by a modification in the noun or verb of the consonantal sound at the end, which is hard in the noun and soft in the verb."

He adds that very often this change of voice is recorded in spelling (eg cloth, clothe; bath, bathe; etc) and that this confirms "the fact that the distinguishing of the parts of speech by change of sound is very common".