Exported videos appear desaturated.

I'll assume from your comments that you're on a Mac, with a 'standard' Retina monitor. As noted in my reply you refer to, the issue is very simple.

The ENTIRE world, outside of Apple, uses Rec.709 video standards including the Bt.1886 appended specs. This means that a display transform is used in the monitor of roughly gamma 2.4 is used.

APPLE ... bless it's littel pea-picking heart! ... for some bizarre unknown reason chose to set the ColorSync utility in their systems to apply the camera transform function of Rec.709 ... approxiamately gamma 1.96.

This is applied to the image in the display, not to the file data itself.

You can't display the same file data with two different display gammas and get the same 'look' to the image. 

Apple Reference Modes

If you have Reference Modes available in your Mac, you can set the screen to HDTV, and then you  will get the ful-on correct Rec.709 video standard applied.

So some Macs will show the image with the odd "lighter" gamma 1.96, some will show the image with the full gamma 2.4 display transform.

Which means there will be different views of any video file even between Macs!

YOUR OPTIONS!

Premiere's new and spiffy color management controls are as of version 24.x in the Lumetri panel's Settings tab. So go there.

Set Display Color Managment to on, unless you know enough about technical CM and how your system is setup, calibrated, and profiled! to know you don't need this. If you don't know what a profile is, you need the DCM set on!

Next thing to check are a pair of settings that are interactive with each other but in different parts of this panel.

Autodetect Log in the Project, and auto-tonemap in the Sequence, set BOTH to on. This will match nearly all HDR forms to an SDR/Rec.709 sequence, or the other way also.

Next, make sure your CM settings match. If the auto settings mentioned above are on, set your Sequence to the color space you want to work in, and your display to the gamma you wish to use, and of course, ONLY use export presets that match the sequence color space.

Display Gamma Options: pick your poison point ...

For Mac users with standard Retina monitors who do not deliver to professional services!

You may be more comfortable setting the Display gamma option to QuickTime/1.96. The image inside Premiere and outside on your Mac, with any app that allows ColorSync to control the image. will be the same.

As the image you see is displayed brighter than normal Rec.709, you will naturally grade it to look darker. But on a non-reference mode Mac, the lighter view is what will be seen.

This would include QuickTime player, and Chrome and Safari browsers. This would not include VLC player, or Firefox browser, which will probably use display gamma of 2.4.

Poison point:  all non-Mac users, and all Macs using reference modes, will display that file at gamma 2.4. Or at the least (for some computers) gamma 2.2. And it will now be a very, very dark image.

Next option: "web gamma 2.2"

This is sorta in-between right? As the computer displays the file with a bit darker shadow area than with gamma 1.96, you will grade it a bit lighter in the shadows.

I do know some that use this as then when shown on a standard Mac, the file is displayed a bit light,  but not quite as light as if the gamma 2.4 setting had been used during grading. And it will be a bit dark on full-on Rec.709 systems, but not too bad.

Poison point: may not be totally accurate anywhere, but oh well what the heck ... (no two screens every show the same identical image anyway ...  )

FINAL Option: Stay to the Standard.

This is done by most of the colorists I know, even though they are  nearly all Mac users. Did you know, all pro produced media, whether streaming or broadcast, is graded for Rec.709 if it isn't full HDR? You're not seeing it as graded anyway!

Yup, that's the mess Apple gave us.

It all has to do with color management:

Everything has to be the same color profile.

Tone mapping in Premiere Pro

Color management in Premiere Pro

ugh that last part supposed to say

"project monitor (right) and the exported clip in quicktime (LEFT)"

just want to chime to say that I did find the following linked thread from a year ago, which included Neils remarkably thorough reply.

https://community.adobe.com/t5/premiere-pro-discussions/desaturated-video-when-exported/m-p/13468555#M447932

And while much of the color science info he discussed admittedly kind of went over my head, I did seem to gather that on the whole the issue seems to lie with a lack of a standard across different systems for display. I suppose since that post is 1 year old at this point, hopefully there is some new light that can be shed on the issue, and best steps for addressing it. Its infuriating to color correct an entire project according to the project panel only for the exported clip to lose its luster, sometimes very to an unsettling degree.

Here is another screenshot showing a different part of the clip, which shows a substantial difference between the project monitor (right) and the exported clip in quicktime (right)