The story
Should is the past tense of shall. In Old English it was sċolde, from sċulan, "to owe, to have to" — a sense of obligation that the word still carries. Middle English wrote it scholde, and the L was pronounced, as it still is in German sollte.
Would has the same story: Old English wolde, past tense of will, with a real L. Both are tiny, extremely common words that usually sit in an unstressed position in the sentence, and weak syllables lose consonants. The L wore away, probably during or after the early modern period, and only the spelling kept it.
Their real L then spread to a word that never had one: could, which English writers respelled in the 1500s to match. So should and would have lost Ls, and could has a borrowed one.
