This is a little like what happened to me, I posted a question here relating to notes that had gone missing too.
It seems that if you take a note while offline, when you sync with icloud it will sometimes without warning delete notes it does not know about :( Luckily in my case since I had made the note on my mac it created a recovery folder for me.
Perhaps your phone may have done a similar thing? If you open notes and hit the 'accounts' button at the top, is there a 'recovered items' folder?
I realized I had the answer at my fingertips, si I'll submit an answer to my own question.
An application I use for file transfers, but sometimes overlook, called iMazing, offers native Notes conversion and backup.
There might be other apps that do this, but this is what I did and how I solved the issue in less than 30 seconds.
1-Link the iPhone with the Notes to your computer running iMazing
2-Launch iMazing
3-Select the iPhone in the list of devices, and click on the "Notes" shortcut in the left-side panel
4-A list of notes will show up. In the Finder, create a folder to receive the backed-up notes, then go back to iMazing, and Select them all [cmd+a]
5-Drag and drop them onto the Finder's folder previously created
The notes will all be converted to .txt and saved as a local file.
It's a great app, what really sucks about it is that you have to pay for it but other than that it does the task of exporting Notes quite well.
Before that, I used to use IExplorer, which worked well too.
I don't recommend using the app "Notes Explorer", unfortunately, it's a free program, so I would have loved for it to be my first answer, but when I tried it it exported Notes in a sort of HTML format with large text blocks and
code snippets everywhere.
There are other apps that claim to do this, at least one other which is called Syncios, which I haven't tried. If you know one that's good, feel free to recommend it in the comments section.
Also, iOS 9 now supports images in Notes, so this answer might be edited depending on what solution will work the best with this new format.
Best Answer
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.