The verb is "accidentally", actually!
If the "correct" sentence is "I accidentally created a scheme", then I think it would be "accidentallied". In excising the verb, shouldn't one carry its tense over to the adverb?
If the "correct" sentence is "I created an accidental scheme", then it would be "acidentalled".
But verbing the adjective doesn't have the same rhythm as verbing the adverb, in this case. And rhythm is the most important rule of English. The indefinite article needs to fall on a low note, and the adverb form is flexible enough to end on a high note to give a lilting affirmation, whereas the adjective form is not.
This is an odd observation. All languages have derivation. ( https://en.wikipedia.org/wiki/Morphological_derivation )
English has zero-derivation of verbs [that is, the form of the derived verb is identical to the form of the word from which the verb was derived], and this construction is highly productive; many people have remarked on it. But there's nothing similar for nouns in English, so I don't see why you're listing nouns as parallel to verbs.
The meme expression "I accidentally [missing verb] [something]" does not obey the rules of English, as you can easily tell by the responses saying "you accidentally did what?"
If you really meant that parenthetical, you might be interested in Chinese, where academics sometimes get into arguments over what part of speech a given example of a word should be considered to belong to. There is very little inflection, which makes the confusion possible.
Cultural differences do crop up. The Romans mostly believed that if you wanted to talk about philosophy, you had to do it in Greek; there is a speech preserved from Cicero in which he complains about this belief and gives several examples of how it's possible to say the same things in Latin by using parallel derivational mechanisms. Construction of new words from roots was common in Greek and unusual in Latin.