I recently read somewhere that with the TES Construction Set you can directly change the number values of all of the facial sliders found in the CharGen, but I think the place I read that mentioned it was only possible with NPCs. Is there any way I can use the CS to input the numerical values of these sliders for my player character? If so, how would I do it?

Well, just do it as described wherever you read it and create this NPC first, in its own ESP. Then you can use Wrye Bash to copy the face of this NPC to the player in one of your save games. Copying the faces is a simple task of a few clicks and should be described in detail in the manual.

One thing that is possible, if memory serves (been a while), is to go to Actors-->NPC-->Imperial-->Male-->Player.
This is the default Player NPC who appears when you first start the game. You can definitely change the default Race, so you should be able to change their hair/eyes/other features-- save it as a .esp file, and you can always start with that NPC appearance.

The only issue is you will not have access to Mod-added things such as new hairstyles/eye colours.