I think it’s punctual because an attempt in my view is a specific action and not a process, but I’d appreciate your thoughts on it.
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It may be either. I quote from grammaticalfeatures.net:
To attempt a field goal in (US) football is punctual, and indeed semelfactive, and may be, in the course of a game, iterative; to attempt to build a consensus behind a policy is durative.
The punctual/durative contrast should not be confused with another pair of aspectual terms which I believe are in play here (same source):
Note that this author ascribes aspect to situations and the utterances which describe them, not to verbs. In my opinion this is the most convenient way to treat not only aspect but tense and mood in English. My opinion, however, is not expert; for a more authoritative treatment I refer you to the link above and to this ELU question and its answers, particularly that by Drew Ward. His CALLE blog has a valuable series of essays on Time & Language which is still in process.
I attempted for nearly five minutes to perceive it as a punctual verb but finally gave up.