No, it should not be used this way. As you pointed out, manifest is a verb meaning display or show. Synonyms are: display, show, exhibit, demonstrate, betray, present, reveal.
Substituting two synonyms, does your sentence make sense to you?
We would quickly learn that obstacles can reveal in any neighborhood.
We would quickly learn that obstacles can display in any neighborhood.
They do not make sense to me. The verb manifest is a transitive verb that needs a direct object. Something usually manifests something else.
Subsequent arguments and destructive forces typically manifest themselves downstream.
Perhaps you want the word manifested. Like revealed and shown, obstacles can be manifested.
We would quickly learn that obstacles can be revealed/be manifested in any neighborhood.
Even though this is a grammatically correct, I would not state something in this manner.