*As far as I am/was aware, AOL emails are only ever openable via an AOL browser.*
I think that Brocken meant that the locally stored Personal Filing Cabinet (organize) can only be accessed via the proprietary AOL Desktop software. However, I have had some (partial) success using this freeware
PFC Viewer Download
I have just tried logging into my_screenname_aol_com and then marking a random selection of Incoming e-mails (going back a couple of years) as unread. Having done that I logged-off and closed Firefox then opened AOL Desktop 9.6 Then I set up and ran Automatic AOL, successfully downloading the e-mails just marked 'unread' in one operation. The date ordering may be a little haphazard.
As for Sent e-mails, I think the advice you received from AOL Mail (now completely separate from AOL UK/TalkTalk) was probably as good as you're going to get. Maybe you could try forwarding them to yourself and then running Automatic AOL again (or forwarding to one of your 6 other available Screen Names thereby keeping the Sent e-mails separate from the Incoming e-mails).
As has been noted here before, if you ever decide to leave AOL you will still be able to use AOL Mail (using any major browser on any OS) and the AOL Windows Desktop ... your Screen Names will remain active. Make things a little easier by creating your full complement of 7 Screen Names before terminating an account (still an ordeal requiring ongoing vigilance despite improvements).
Pippa's Parties E&OE