In terms of your query, filters correct the color shift towards blue or green that occur underwater.
Post-capture white balance creates a global color shift. The most common result is that this drains away the blues or greens (as the software shifts the spectrum toward the red end). This leaves them looking very washed out.
Shooting in RAW captures all the sensor information, but given that there is little information in the red end of the spectrum at depth, it cannot capture information that simply isn't there.
By reducing the amount of blue/green light that the RAW image is capturing by using a filter, this allows the camera to white balance in a way that is more natural looking and retains the deep blue and green colors of the water column.
Most images have some sort of post capture work done on them. The point is that shooting with a filter on some types of image will give you better results post capture.
In terms of whether you chose to believe the information presented in the shared link, that is up to you. I guess the many underwater photographers that successfully use filters to enhance their images regularly will continue to do so, and if you don't want to use them that is up to you.
BTW, if you put "filter" into the search button above, or Google "Wetpixel filter" or similar, you will end up with over 100 results, some of which would give you more understanding about filters...just saying