How can I extract locally stored Notes from iOS without enabling the cloud? A family member has been using an iPhone since around 2012 and regularly uses the Notes feature to store various things to remember. At some point, syncing of notes was no longer supported by Apple as they decided that everything should be on iCloud. Said family member does not want anything on the cloud.

A few upgrades of the OS later and we're now in a situation that they have these multiple notes on the device, but are ONLY accessible on the device. Though I presume they're all getting backed up as part of the phone's backup process on iTunes, I'm imagining that they're stored in some binary format (PostgreSQL?) and not easily extracted if they decide one day to switch to Android. The only way out, it appears, is to COPY every note to either an email or to an iCloud note which neither is very appealing.

Is there any way to do this? Preferably, it would be ideal if the notes would sync with their laptop's Notes application like it used to a few years ago, but I'm assuming that Apple won't change that back.
This is a frustrating problem, and many folks are not happy with Apple's decisions regarding this direction of iCloud only syncing.

While you can use Airdrop to copy individual notes from iPhone to the notes app on the Mac, that entails copying each note which you have already said is onerous (I would agree if there are many notes). The advantage this process has is that it's free, and it moves the notes into the Notes app on the Mac.

Since this iPhone isn't synced anywhere in the cloud, there are several commercial apps that you can run that allow you to control your sync options with your iPhone. These have been made popular by Apple restricting what you can and cannot do between an iOS device and a computer.

Two that I have tested, and offer free trials, are SyncIOS and iMazing.

Both of these are generally well reviewed and easy to use. Both have limitations of their free trials and cost around $35 USD to unlock their full potential.

But even these programs don't actually support syncing of notes between Mac and iPhone. You can see and read the existing notes on the phone, but you have to export them to the desktop and then manually import them into Notes. Note here, SyncIOS allows you to actually create new notes from within the app on the iOS device!

So Notes, images, music and even voicemails are easily accessible, and you can back them up as well on your local desktop. Both apps come in Mac and PC versions and both require a USB connection to the iPhone or iPad.

For my money, iMazing is a more polished, feature rich app and it can easily access existing backups from iPhones or iPads. But either of these fine apps will meet the basic needs you mention in your post.