Consider: what is the voice of "this is my first time being seen" and
"I am excited to be seen"?
I've considered and these sentences are also active. (Also, they state different things so in terms of voice, there is no point comparing them with the original sentences). The syntactic active/passive contrast in English is all about turning the head verb of the predicate into be + past participial form of that same verb. (There are a couple verbs that express the same semantic contrast using -ing forms, and also there are "bare passives" and "get passives", but centrally active/passive contrast is expressed by this simple transformation.)
So, we need to identify the clause that is headed by the verb that undergoes this transformation. It obviously cannot be the main verb "be", so we need to look into the subordinate clause. In the first example, it is the participial clause headed by the verb "seeing". The aspect information has to be transferred to the first verb, which is now the passive auxiliary "be" and we have : seeing --> being seen. In the next step we'll have to switch the roles of the participants, which is the whole purpose of this transformation. In this case the object of the non-finite clause "Disneyland" becomes the subject of the corresponding passive clause: "Disneyland's first time being seen". This clause stands in voice contrast with the active "My first time seeing Disneyland".
This doesn't affect the voice status of the main clause: "This was my first time seeing Disneyland" is active and so is "This was Disneyland's first time being seen"
The genitive "my" functions both as the determiner in the noun phrase, and as the understood subject of the participial clause. The voice contrast it enters makes a good case for analyzing it as the ultimate subject of the non-finite clause.