Post-functions are only applied on transitions, which changes the status of an issue. Leaving a comment is a separate type of action/event in Jira, and post-functions are never fired when leaving a comment.
If you don't want to use an add-on, I would just instruct who ever the first responder is to click the "Assign to Me" link near the Assignee field.
If you're open to using add-ons and if you want the first commenter to be assigned to the issue - the only way I know to do this is using Script Runner for Jira Server and create a scripted listener for the "Issue Commented" event (I think Script Runner for Cloud can do the same thing). If the issue's Assignee field is set to Unassigned, then assign to the current user (because that's who just made the comment).
Re: "Never assign the ticket to a particular user" - I'm not sure what you mean by this. Are you saying you don't want to be able to edit the assignee field after it's set for the first time? That removes a lot of flexibility from being able to change the assignee for any reason (someone's on vacation, leaves the company, etc.), and I wouldn't recommend it.
There are a couple of options here, I think:
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