In order to make any progress, you will need Fusion Explorer. Grab the download link in the description of the video.
Once you download it, open up the .zip file and drag the folder out onto your Desktop.

It is also recommended you make a backup of your data.pak. You can do this by browsing the local files (see image; right click on the game to open up this dialog).
Then go to "build/data_pc/". You should see the data.pak right there. Copy-paste it in the folder so you have the original and a copy.
If you don't and you mess up really badly, you will have to go to the game properties and verify the integrity of the files. (It's not a big deal if you mess up since it'll just take time redownloading the 500MB.)

Once you're done doing that, go back to the main Trials Fusion folder to begin the next steps.

Go into your Fusion Explorer folder. Click on View at the top of your File Explorer window, then check the box for "File name extensions"; we'll need this for later. Now open up the exe file within the Fusion Explorer folder. Since you still have the File Explorer open from earlier in your Trials Fusion folder, copy the path at the top. It'll look something like "C:/Program Files (x86)/Steam/steamapps/common/TrialsFusion".

In Fusion Explorer, click File in the top left, then click Open. It'll bring up a File Explorer window. Paste the url into the top to get yourself instantly in the Trials Fusion folder. Now go to build/data_pc/ and select data.pak by single clicking on it. Then click Open. Double click data.pak in Fusion Explorer and you should get something that looks like this.


Double click on "soundsystem", then double click on "win". We should have a long list of files that end in extensions like .fsb and .lst.

Multiple operations on your data.pak can corrupt it. It is HIGHLY recommended to get all your modifications ready (eg these mods, any customization mods) and then SAVE ONLY ONCE.

This section will remove all the music from Trials Fusion. You have been warned.

Create a new file (.txt. file is fine) on your Desktop (or somewhere convenient). Rename the file (including file extension) to TrialsFusion_Music.fev. You should get a prompt that looks like this:
If you didn't get this prompt, you are not including the file extension (.txt). Once it's renamed, you can verify you did it correctly by double clicking on the file. Windows should ask you "How do you want to open this file?". You can click anywhere else to get this prompt to go away.

Go back within Fusion Explorer and right click on the file "TrialsFusion_Music.Fev". It'll be near the bottom of the "win" folder.
Click on Import. Use the File Explorer popup to navigate to where the file we just made is. Select it and click Open. In Fusion Explorer, do File -> Save to save our changes. Right click on the TrialsFusion_Music.fev within Fusion Explorer and click Info. The size of our file should read 0 bytes; that's how we know the changes were successful.

Create a new file (.txt. file is fine) on your Desktop (or somewhere convenient). Rename the file (including file extension) to TF_AudioLogos.fsb. You should get a prompt that looks like this:
If you didn't get this prompt, you are not including the file extension (.txt). Once it's renamed, you can verify you did it correctly by double clicking on the file. Windows should ask you "How do you want to open this file?". You can click anywhere else to get this prompt to go away.

Go back within Fusion Explorer and right click on the file "TF_AudioLogos.fsb". It'll be near the top of the "win" folder.
Click on Import. Use the File Explorer popup to navigate to where the file we just made is. The popup should be asking for a .fsb file; if it isn't, you may need to restart Fusion Explorer. Select it and click Open. In Fusion Explorer, do File -> Save to save our changes. Right click on the TF_AudioLogos.fsb within Fusion Explorer and click Info. The size of our file should read 0 bytes; that's how we know the changes were successful.

Optional
Go within the Trials Fusion folder, then go to "datapack\flash\video". You should see a bunch of folders and then 3 .bik and 1 .ogv file. You can delete all the files; do not delete any folders.

Note: You will still have to wait out the length of the videos, but at least the audio and visuals will be gone.

Thanks to Scoped in the Trials Community Discord for giving me tips on how to do this.
Thanks to Jake QK for creating the Fusion Explorer tool that makes this possible.