I don't think it has to do with them being verbs, and for the most part, I also don't think it has to do with the etymology. I think it mainly has to do with the stress patterns of these words.
The tendency I see is that the letter L is generally doubled after a stressed short vowel, and not doubled after an unstressed vowel. Historically, there has been a lot of variation in the spelling of words like this. Based on etymologies that I have looked up, I would say there is little to no etymological significance to the use of a single L vs. a double L at the ends of words. It's just an oddity of our modern spelling system. (Another relevant question about double L: Why is there a double "ll" in "bell"?)
Here are some actual examples: mill (as you've found out, it comes from Old English mylen), small (comes from Old English smæl), skill (comes from Middle English skele, scele, skile, skyle, from Old Norse skil), dill (from Old English dili, dile, dyle). On the other hand, metal comes from Latin metallum with a double L (although the French spelling metal may have also played a role, as French seems to have been an intermediary in the transmission of this word). I have some more examples of words like metal in my answer here: Adding an L when appending an -ium suffix to a word? (Metallium vs. Metalium)
Monosyllabic content words always end in a stressed syllable, and polysyllabic verbs are more likely than nouns to end in a stressed syllable. These two factors are probably why it seemed to you that this was related to part of speech.
There are many nouns that end in "-ill," especially monosyllables such as bill, hill, sill, and frill. Adjectives such as still can also have this spelling.
There are also verbs that end in "-il." Aside from bedevil, there are cavil, imperil, stencil, fulfil (British spelling; Americans use fulfill). However, it does seem to be true that fewer verbs than nouns end in "-il."
What is the etymology of words that end in -il?
It's worth noting that there aren't that a huge amount of words in total that end in -(consonant)-il."
As you've found, many of them are derived from Latin nouns with stems ending in "-ill-" (for Latin words ending -illus, -illa and -illum, the suffixes -us, -a and -um indicate gender and other inflectional categories; they are not part of the stem). When Latin nouns are taken into English, inflectional endings such as -us are often simply removed, which explains how we get English nouns ending -ill or -il.
Latin verbs, however, are usually borrowed in a more complicated fashion. For complicated historical reasons, Latinate verbs in English are generally derived from the past participle form of the Latin verb. We do truncate the gender/number suffix -us/-a/-um from that, but Latin past participles also generally have a dental suffix (usually t) that is retained in the derived English verb. So for example, the Latin verb inficio corresponds to the English verb infect. In regular verbs of the first declension (the most common type) this dental suffix is preceded by the vowel a. So Latin verbs with stems ending in -ill- have generally been borrowed as English verbs ending in -illate. (For example, scintillate comes from Latin scintillare). So this might be a contributing factor to the relative rarity of verbs ending in "-il."
There are various other sources for words ending with "-il." In most cases, the modern spelling with "il" is fairly straightforward based on the etymology.
However, there are a handful of words that seem to have acquired the spelling "-il" for unclear reasons. Here's what I mean: devil (Old English deofol), evil (from Old English yfel), weevil (Old English wifel), stencil (earlier spelled stancel; interestingly, the Oxford English dictionary says that the noun is derived from the verb). I don't know how to explain these.