We know that it is OK to start a sentence with a conjunction, but not OK to have incomplete sentences. This seems to me like a contradiction. How can a sentence that starts with a conjunction be a complete sentence? By its nature, a conjunction links to a previous sentence. Therefore, a sentence that starts with a conjunction does not form a complete thought, and is therefore incomplete.
Start sentences with a conjunction (Grammarly):
There is nothing wrong with starting sentences with “and,” “but,” or other similar conjunctions. You may, however, encounter people who mistakenly believe that starting a sentence with a conjunction is an error, so consider your audience when deciding to structure your sentences this way.
Avoid incomplete sentences (Grammarly):
No one escapes high school English without being penalized for writing the odd sentence fragment, but not everyone remembers what they are and how to fix them. Put simply, a sentence fragment is a clause that falls short of true sentencehood because it is missing one of three critical components: a subject, a verb, and a complete thought.
Fragments in speech vs Fragments in novels? does not quite answer the question, since it doesn't resolve why it is fine to start a sentence with a conjunction even if conjunctions create a fragment.