Hi Brandnewuserid
Welcome to this Apple (user-to-user) Community !
Re: "Is there a way to drag Apple mail items to local folders?"
Wondering which MacOS version is installed on your Mac mini.
Eg: Am using MacOS 13.7.6 "Ventura" at present, and have always dragged emails from the "Mail" Inbox to the e-mail folders (called Mailboxes in the Mail app) in the sidebar on the left. I've created these named mailboxes in "On my Mac" in the Mail app itself, (ie: look underneath the default Inbox, Sent, Draft mailboxes at the top).
Eg: Here's a link to Apple Support's article for MacOS 15 "Sequoia"
NB: For accurate info: If you are using an earlier MacOS version, you can choose that at the top of the article::
See how to: Move or copy emails between mailboxes in Mail on Mac - Apple Support
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Re: "Is this just not possible with Apple Mail because it is web based only, and staying that way"
The Mail App on Mac is in the dock, Any eMail Accounts can be added to it, so your emails from your providers (Yahoo GMail etc.) appear in the Inbox.

If you want to add your Mail accounts to Apple Mail : See: Get started with Mail on Mac - Apple Support
You will likely need to give your email provider permission to allow Mail to access your emails, and to choose what happens after the emails download to Mac .... eg: whether or not each email is also kept in the account in provider's website. If needed: Most email providers have a "Help" or Support" section, to help with permission and other settings.
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Re: "I cannot get used to the idea of not being able to store selected emails locally"
For me, the Mail app makes it easy to receive, answer and store my eMails from 3 web-based accounts directly on my Mac. More info. / tips / support / contact options etc. Mail - Official Apple Support
Resource: MacOS Sequoia (or choose OS at top of articles) : Mail User Guide for Mac - Apple Support
All the best :-)