John was happy about being accepted as a team member.
In one sense, John is an accepted team member, so it could be an adjective.
In an other sense, John was accepted by someone, or by the team or coach. In this sense being accepted is a passive clause. But I suspect that the passive is not really established, because as a verb it still lacks attribution to an agent, and even if it did, John is the subject of the clause, the agent of the action is not.
(1) Parsing the sentence: John was happy about something.
John was happy about being accepted as a team member. S|V|SC(AJP+(pp):happy+(pp):accepted)
So in this parse, the SC is complex. The core SC (happy) is modified by the pp about being accepted, which is complex in itself because it takes a clause (copular verb + adjective) as an object, and accepted is also modified by another pp. Both of these pp are integral to the SC because they are tied to the core AJP happy; there is a chain of semantic dependency.
By core I mean any part of the sentence that is required in terms of syntax or meaning. If you remove a core part, the sentence breaks. The rest is just added, non-essential information.
So, in this parse accepted is an adjective.
(2) Parsing the sentence: John was happy about something.
In this parse the focus in on the clause being accepted where being accepted is taken by some as a passive construction, simply because it could take an agent attribution with the pp by the coach.
John was happy about being accepted by the coach as a team member.
But if this is a passive form then it should have an equivalent active form as well, but it doesn't because John is the subject of the clause being accepted. So even with this attribution of agency the participle is still an adjective, or at least it is not a passive verb.
I would have to say that accepted is an adjective, by zero-derivation, but I'm not confident in that, given what a newbie I am.