So where is there any evidence that telekinesis has existed with any real proof?
There are many mages of legend and modern fiction- Gandalf and Merlin, for example- who used telekinetic powers. Its something that isn't unique to psionics.
In fact, most common fictional psionic powers have some kind of analog in fictional magic powers.
Pyrokinesis? Fireballs.
Using TK to stop someone's heart? Finger of Death.
Levitate? Fly.
TK shield? Walls and shields of arcane energy.
Also please define "general fantasy". What I see has something much more closely related to a wizard than a psionicist.
So what exactly is giving the impression that psionics is closer to how magic works?
D&D was taken from, Tolkien and other like materials, mythology, so why should it not fit within those ideas but somewhere else?
1) For a pretty full discussion on the distinction between fantasy and sci-fi elements like psionics, check out this thread:
http://www.enworld.org/forum/general-rpg-discussion/132875-genre-conventions-what-fantasy.html
2) Psionics (and Sorcerers) works more like traditional magic in that it is entirely unlike Vancian "fire and forget" magic that D&D Wizards use. A traditional mage could typically do whatever arcane stunt he knew as long as he had the power- mana, willpower, souls in a jar- to do so. This is something Psionic PCs, Warlocks, Sorcerers and the like all do better than the Wizard, and none does it better than the first 2. The psionic resource of power points iis mechanically distinguishable from mana only in terminology. In addition, the old idea of mana is nearly indistinguishable from the Force which is manipulated by mitichlorians (OH, give me a break, GL!)- the distinction is that the wizard directly manipulates mana, the jedi or sith manipulates the force by way of the mitichlorians.
3) Your source-list for D&D is woefully incomplete. In addition, it was shaped by many sci-fi sources, like Jack Vance's
Dying Earth (from which we get the D&D magic system, IOUN stones, and other things), or the Eternal Champion cycle which uses both magic and science-fiction elements. Some of the critters not culled from mythology came from sci fi literature and movies. 1Ed D&D had psionics, robots, blasters starships, and even a fantasy take on mecha.