"Gone" is the past participle of "go" just like "dispatched" is past participle of "dispatch".
Syntactically, "be gone"
is does have a structure that looks like the passive of "go", just like "be taken" is the passive of "take", and "be used" is the passive of "use". But "be gone" has no equivalent active voice, and can't be followed by "by someone", unlike true passive voice verbs. It is used as an adjective, and is sometimes called false passive.
"He has been gone" is present perfect tense, so "gone" used as an adjective is in past participle.
"Gone" can be used as an adjective, and so can other verbs in passive voice ("The message is long" / "the message is deleted"). The separation between passive-voice verbs and stative-passive adjectives is intricate, and there is sometimes ambiguity, especially when used in a past tense ("was deleted"). This can be solved by using "get" instead of "is" to unambiguously signal passive voice: "the message got deleted".
See here for more discussion and examples of present perfect passive.