Anathematically said:
Tarot didn't even exist during the time of the pythagorean elements.
I am no historian, but I would say that this is more an argument in favor of the suits being based on the elements than not; the knowledge was out there and maybe even in vogue given the religious leanings of art in the middle ages (a divine trinity + earth).
Also I think there is evidence of the playing card suits coming from the east which also had an elemental structure.
Ultimately, does it even matter what the first person who did it had in mind? It fits a spiritual world view so well....
Also, I know there is always a lot of confusion regarding the Fire/Wand, Air/Sword attribution. Here is my take on it.
From what I have observed the Wands/Fire is the most usual corespondence (and the one I prefer) and it is the one favored by most occultists.
Sword/Fire, Wand/Air is usually favored by wiccans and neo-pagans. I think that is because wiccans have different magickal tools they use to symbolize the 4 elements like in the creation of an alter, and they use daggers to symbolize fire and a wand for air (if I remember correctly!). You can see how this would cause some confusion! And those with that spiritual lean have incorporated that into their decks, some of which are popular.
My ideas about them are this:
Fire is creativity and the Will, the masculine aspect of duality, yang. It is represented by a Wand because of its phallic nature (resemblance to a penis) , its connection to the life-force, notice how many wands are illustrated with green leaves, and of course the image of a wizard or magician using a wand as an extension and amplifier of his Will. The Wand is associated with the 4 Kings. The Wands/Fires counterpart is the Cup/Water, the receiving womb who is also symbolized by the 4 Queens.
Air represents the intellect. The ability to examine and decide. It is cutting and sharp, think of the Queen of Swords. It is represented by the 4 Knights. I have heard it compared to the warrior class as opposed to the ruling class of the Kings. One of the reasons I really like a sword for Air is because of its dual nature like the 2 edges of a blade. On the Tree of life it is Tipareth, a pivot point between God and the manifested earth and I picture it moving between t he two like a lightning strike, which is the equalization between the earth and clouds which have a positive and negative charge. Very active, concise and even extreme. I also like to think of Fire and Water being the first duality, the yin and the yang, the King and Queen, and if they have sex they have a child with a body (earth or pentacles) and a soul (Swords/Air). So pentacles/Earth is like a second generation Yin/feminine and Swords/Air a second generation Yang/Masculine.
The elements in the Thoth's court cards are different, mind you.
I could go on for hours about this but will let it be at that!