It have nothing to do with servers. They have nothing to do with graphics, they just tell the clients what parts they are supposed to use. Everything you see on your monitor, other than numbers and written messages, is dealt with by your computer. Everything. Server manages the invisible things, the mechanics of the game. Server needs not generate graphics and, in fact, if it did, there would be no such thing as minimum computer specifications to run FFXIV, other than high-speed broadband internet connection (in reality there would be, but something that a 20 year old computer from bottom shelve would mostly meet).
The "spaghetti code" is also not exactly responsible here. It doesn't matter how messed up the writing is, it does matter what the engine allows. If the engine does not support higher quality hair physics, you won't have higher quality hair physics.
This is actually false when we consider the clipping. Remember, it's not the hair physics that are responsible for clipping. A game does not need any hair physics beyond the basic "they can move, so they can bend". It's about collision detection. And that requires proper 3D models.
What does create the clipping is the fact that...there's actually no tangible 3D models in this game. There is no collision detection for characters at all. It's just a bunch of textures upon textures, jumbled up. I bet there is a bare-bone system of sorts to manage emotes and attack animations, but that only applies on the "body", and that includes the body parts of the clothes. Everything else is completely ignored and "stiff", able to pass by anything and everything. If it was made "tangible", even only towards the rest of the clothes and hair, the clipping would disappear (outside of occasional bug).
That is easier said than done however as the amount of work necessary would be significant. Not only would one need to make every part of the clothes have some sort of physical traits, but there also needs to be a code that decides what to do when two items collide. If there's a long collar to the robe, does the long hair fall inside or outside? When you move the head, does the hair move around the collar or is pulled up? If you have earrings, how they move when they touch that collar? All of this requires a great deal of programming and menial work that Square Enix SHOULD be able to afford...but it would require a complete overhaul of the way the game graphics work. Basically, it probably would require remaking everything in the game from concept phase + all the programing necessary to even implement such systems. That's why it's a feature that either is there from day one...or not at all.
All of that however is possible with significantly less computer power than you seem to think. There are many years of games with lesser clipping issues from before "realistic hair physics" were even a thing. That's because it's the exact same technology that prevents your character from walking through a wall or falling down the earth, with only basic physics for how the materials interact (because unlike with a wall or floor...the game cannot just forbid the movement when collision occurs, as that bug the graphics at best or crash the game at worst the moment you'd use any combination of clothes and/or emoticons that cause such collisions).
And again, this is not necessary for preventing clipping. Those are high-end hair physics that allow the hair to move realistically, but it would not even stop clipping in the first place. It would only solve the problem of how the hair would be interacting with the clothes once clipping would be removed.
For removing clipping, using the current hair physics is more than enough. It's a matter of letting the game detect the collision and having the game decide on how to deal with it (by properly bending certain parts, in this case hair, so that they are over the other objects they collide with rather than go straight through them). As above, it's easier said than done, but resource-wise, it is far less demanding than advanced hair physics.
All in all, this game will never get rid of clipping properly, unless the graphic engine will be completely changed. Yoshida said that there is no chance of that for at least a decade...so yeah.