Hello, @Matthew Rosier
Welcome to the Microsoft Q&A platform!
Unfortunately, the Exchange Administration Center (EAC) itself does not directly support exporting mailbox content to .PST files. The purpose of exporting mailboxes using EAC is more for managing mailboxes than exporting mailbox contents.
To export emails from mailboxes, you will need to use the Microsoft Purview eDiscovery tool as you initially mentioned. You can easily achieve this according to How to Export an Office 365 Mailbox to PST - [Step-by-Step].
(Please Note: Since the web site is not hosted by Microsoft, the link may change without notice. Microsoft does not guarantee the accuracy of this information.)
Meanwhile, if you don't have the need to export the contents of multiple mailboxes at the same time, you can also choose to complete the export operation in Outlook client directly with the help of Export emails, contacts, and calendar items to Outlook using a .pst file - Microsoft Support, which can also help you save a lot of time.
Should you need more help on this, you can feel free to post back.
If the answer is helpful, please click on “Accept answer” as it could help other members of the Microsoft Q&A community who have similar questions and are looking for solutions.
Thank you for your support and understanding.
Best Wishes,
Alex Zhang