Seems odd that a verb with iĝ translates as a transitive verb. I thought the iĝ suffix was invariably intransitive
It's not transitive. That's why you say 'al la klubo', not 'la klubon'.
I don't even know what a transitive verb is... could you point me to a resource that explains transitive verbs and all of the other "kinds"?
When an intransitive verb has a preposition prefix, it may be treated like transitive verb, one that takes the same object as the preposition. This is most useful to make passives or to coordinate verbs.
Ŝi eksciis kaj pripensis tiun fakton. La loko alvenita...
When the verb occurs by itself, it's common to keep the preposition.
Pripensu pri tiu fakto.
Is aligxi intransitive? I thought this should be "aligxi la klubon", or am I mistaken?
One could argue that "aliĝi la klubon" is OK by the preposition-substitution rule, but "aliĝi al la klubo" is what you will hear most Esperanto speakers say.
Edit: it was brought to my attention by one of the course authors that "aliĝi la klubon" used to be a sentence in the course, but has been removed as an error. See this thread.