Hmmm, I was under the impression that email was collected to a schedule.AFter a quick check that would appear to be the case. I'll see if it's supposed to be run on a schedule.
Your description isn't a catchall account. A catchall account is used to receive mail for an address that doesn't have a mailbox on your server. So if you had an email addressed to firstname.lastname@example.org and that didn't exist it would get sent to the catchall account.
What you're describing for getting mail from your ISP account isn't easy (AFAIK) using fetchmail or getmail. They're both meant to get mail from one or multiple external accounts and deliver them to a single mailbox. The mailbox you describe at your ISP is called a multidrop mailbox and are usually not a good idea as fetchmail etc tend to get confused by the headers on those emails.
Having said that you could always set-up the account on your server that mail is delivered to with a filter that forwards a copy to the correct recipient - assuming that there aren't too many.