Did Japanese verbs ever end in consonants?

One of the most distinctive features of the Japanese language is its relatively simple sound system. Ever since Japanese was first written down around 712 AD, Japanese syllables have always begun with at most one consonant, and ended in a vowel — these are called open syllables. In Modern Japanese, there is sometimes an n that looks like a syllable-final consonant (as in Nihon “Japan”), but that n actually functions as a vowel (a syllabic n).
Without syllables ending in consonants, it is strange then, that throughout its recorded history, Japanese has also had verbs stems ending in consonants. These “consonant-stem verbs” take different affixes from “vowel-stem verbs.” For example,
nom–u “to drink” (stem: nom–)
nom–anai “to not drink”
tabe–ru “to eat” (stem: tabe–)
tabe–nai “to not eat”
Stems and affixes usually get their start as independent words, which gradually fuse together over time. But Japanese has no syllables ending in consonants. How did it come to have verb stems ending in consonants?
Here’s a clue: this difference in treatment between stems ending in consonants and stems ending in vowels is even more prevalent Korean, where case markers also differ depending on whether they follow a consonant or a vowel. For example, the nominative case marker is –ga after a vowel (e.g. nae–ga “I”) but –i after a consonant (e.g. saram–i “people”).
This differentiation makes sense in Korean, where syllables can end in both vowels and consonants. But in Japanese, syllables do not end in consonants. The fact that consonant-stem verbs still exist in Japanese, then, appears to suggest that, once upon a time, Japanese used to also allow syllables to end in consonants, as Korean does today.