[Xiaomi Redmi 3 ROMs, Kernels, Recoveries, & Other ] flubba86:
Just an update for anyone else reading this who comes across the same issue.

I fixed my issue and wish to share the solution.

It is not Xiaomi-specific, and maybe not even LineageOS specific.

I believe this problem was caused by me using Titanium Backup to restore a lot of backed-up apps and settings from my wife's old Huawei phone onto to the new phone. And it ended up being a combination of two different problems.

Firstly, the "Media Storage" system app was not seeing any of the ringtones in /system/media/audio/ringtones, but all of the .ogg files did exist in that folder.

I tried:

Booting into Recovery. Wiped Cache partition, and Dalvik/Arc cache. And re-flashed LineageOS.

Used App Info to clear the Data and Cache storage of the "Media Storage" system app.

Used Titanium backup to forcefully uninstall the Media Storage system app, then reinstalled it by reflashing LineageOS again.

None of those things worked, the Media Storage app was still not seeing the files in /system/media/audio/ringtones

I found an app in the Play Store that allows you to force Media Storage to rescan directories. I ran that, but it only scanned in /sdcard directories, not in /system/media/audio.

It was around this time I noticed the second problem. The LineageOS settings App was no longer asking to use Media Storage as a Ringtone picker. It was not even asking which picker I wanted to use.

The current ringtone in Settings was set to "Unknown", and pressing on it to choose a different ringtone did not bring up the option to use Media Storage, it simply brings up the list with a single item "None".

Same with Notification tone and Alarm tone. Pressing on it to choose a different option did not bring up the option to use Media Storage, it simply brings up the list with a single item "None".

I installed CatLog to check for any error messages when opening the Settings ringtone picker. There were lots of errors listed, but one stood out.

Something along the lines of "No applications registered for intent Action RingtonePicker". So the settings app no longer knows that "Media Storage" system app should be used as the ringtone picker, so just shows "None".

Things I tried to fix that.

Booting into Recovery. Wiped Cache partition, and Dalvik/Arc cache. And re-flashed LineageOS.

Used App Info to clear the Data and Cache storage of the "Settings" system app.

Used Titanium backup to forcefully uninstall the Settings system app, then reinstalled it by reflashing LineageOS again.

Cleared the Default Apps list for the Settings app.

Cleared all default apps choices in LineageOS.

None of those worked. Settings app still did not ask me to use Media Storage system app as ringtone picker.

Out of final desperation, I found an alternative ringtone picker in the play store, and installed that, to see if the Settings app would now give me the option to use that.

With the new Ringtone Picker installed, the Settings app does ask me, and gives the option to use "Media Storage" or "Alternate Ringtone Picker".

For the first test I used "Alternate Ringtone Picker", and it has the ability to use Media Storage itself to list the ringtones. Choosing that option, I finally see the fill list of installed Ringtones.

So at some point during my troubleshooting I must've fixed my first problem, and Media Storage app did rescan the /system/media/audio directory and repopulate its list correctly.

After choosing a random ringtone, the settings app no longer shows "unknown" as the selected ringtone. And now when pressing the chooser, I can select "Media Storage" directly and it brings up the list correctly.

So I don't even need to use the Alternate Ringtone Picker anymore, and I uninstalled it.

Im not 100% sure what actions fixed the two different problems, but I believe it was 1) using the Rescan app to force Media Storage app to rescan its directories and repopulate its ringtone contents, and 2) installing a different alternate ringtone picker to force the Settings app to bring back up the option for which ringtone picker app to use.