This is the best way to work around the google family link on a Chromebook. Keep in mind that this will force wipe your device and you will lose files that are not backed up

DISCLAIMER

THIS METHOD MAY VIOLATE GOOGLES TERMS OF SERVICE OR VOID YOUR WARRANTY. THERE IS A SMALL BUT REAL CHANCE OF BRICKING YOUR CHROMEBOOK. PROCEED AT YOUR OWN RISK AND WITH CAUTION. I ASSUME NO RESPONSIBILITY FOR ANY DAMAGE INFLICTED WHETHER PHYSICAL, DIGITAL, OR JUDICIAL RESULTING FROM FOLLOWING THESE STEPS.

Supplies

You will need the following.

A Chromebook with or without Family Link, may not work if you have a school device.
Enough Linux experience to do simple things
Willingness to lose not backed up files

Step 1: Back It All Up

First backup anything you have in your downloads folder etc. to google drive if it isn't already. Also if you are feeling extra cautious and don't want to mess up and brick your Chromebook I would recommend backing up Chrome OS on a USB drive. (Instructions here)

Step 2: Enter Developer Mode

READ CAREFULLY BECAUSE YOU CAN NOT REFERENCE THIS TUTORIAL DURING THIS STEP

Press: ESC+RELOAD+POWER
You will see a screen that looks like the above image.
Press CTRL+D
The system will do some loading and will wipe your local data and enter developer mode. (expect this to take a while.)
Sign back into your google account.
Return to this tutorial

Step 3: Crouton

No, not the salad topping pictured above (no matter how delicious it may be).

This is crouton for ChromeOS, basically, the software package that allows you to run a Linux desktop in front of ChromeOS, therefore, bypassing Family Link screentime limits. You can download it here.

Now press CTRL+ALT+T to open a terminal window and type

shell

now type

sudo install -Dt /usr/local/bin -m 755 ~/Downloads/crouton

That makes the installer executable

the next step is to type:

sudo crouton

you will see a long list of commands, make note of these if you plan to do anything special with crouton.

Step 4: Installing Linux

Now it is time to finally install the Linux environment you wanted!

In CROSH type,

sudo crouton -t xfce

and press enter. For a long time after that, you will see some scrolling text, and eventually, it will prompt you to create a password. Once that is done your text cursor will reappear before the "chronos@localhost". At this stage type the following:

sudo startxfce4

your screen will go black for a few seconds then the XFCE ubuntu-desktop will pop up.

Step 5: Switching Between Them

To switch between Linux and ChromeOS quickly you can press CTRL+ALT+SHIFT+BACK

Step 6: Family Link

Skip this step if all you want is Linux and do not have a Family-Link managed device.

So assuming you have a device that is family link managed then it is likely that a screen-time limit has been set. If you need a way to keep working or playing after that timer runs out then all you have to do is press CTRL+ALT+SHIFT+BACK or CTRL+ALT+SHIFT+FORWARD to use the Linux environment you made with crouton.

Step 7: Using Linux

If you are already a Linux aficionado then you don't need me to tell you how to do stuff and so I won't proceed. However, if you feel stuck or don't know a command then here is a list of good resources.

The crouton README.md has loads of good information presented in a digestible manner.
All the helpful communities on Reddit that are involved in this topic such as r/crouton, r/ubuntu, and r/chromeos
The household beloved www.google.com/
As well as Instructables and relevant discord servers