This concept might fit a bit better in a science fiction game, or a science-fantasy/fiction game masquerading as a traditional high fantasy game. This won't apply to everything perfectly, but since some consider psychic powers as magic by any other name, I'm sure it will work.
Wizards are Psychics. Spells are psychic powers. This works best for spell systems using spell points or “stress points” based on how many spells are cast giving increasing dangerous negatives to rolls. However, Vancian spell slots could also function; simply requiring the fluff that spells are concentrated packets of psychic energy formed in preparation for later, ala a Tulpa.
Telekinesis is your core ability. You can move objects with your mind- Unseen Servant, Levitation, Mage Hand/Floating Disc, Hold Person/Portal and so on. More complex forms of movement, such as to animate a skeleton or controlling wind requires higher levels of power. In this magic system, golems and skeletons would be appropriately mindless automatons, continually animated by some psychic commands and energy. Similarly, giving a mind to something would be a pretty impressive feat, or involve the creator giving up a part of their psychic mindspace to the creation, similar to spending XP.
Pyrokinesis and Electrokinesis explain elemental spells. Cold spells aren't as easily explained, unless you just count it as really focused telekinesis that stops atoms in place.
Summoning spells are in a weird place. Summoning could be either literally creating new life through a combination of telekinesis and more abstract energy- that's why you need a sacrificial virgin to summon a demon, her life energy and body is drained to form the new living creature before you. Or it could be a totally psychic phenomena like a Eregore. It only exists in creatures minds, which is why demons try so hard to make themselves appealing and get people to do their bidding, manipulating people to giving it power or fulfilling its mission. Summoning could also go with a more “New Age” model of cosmology similar to some New Age Religions- Demons exist as a “lower vibration” form of being that your psychic energy can elevate to your plane. Gods and angels exist on a “higher vibration”, all of which require intense mental effort to have them manifest here.
All Illusion & Mind-Domination spells are telepathy. Psychic false sensations and personalities existing only in the mind. This wouldn't work for semi-real illusions, unless the illusions can do damage by hijacking the creature's body internally and causing real wounds to rip open, but at that point you may be overstepping what makes Psychic feel “Psychic”
Healing works by tricking a body into going into super regeneration mode, or by actually influencing life energy. Based on a “vibration” nuAge model of cosmology, you could also say healing is restoring things to a platonic ideal or granting positive energy in droves. It could also just be telepathy based- only masking the pain a person feels and acting as 'Grit' or abstract Hit Points.
Teleportation isn't a common power of psychic characters, but could be used as is. Instant transmission of matter across space, or by 'folding' space. If you use a psychic otherworld or a magical collective conciseness connecting all beings, then this could also work but implies that you can only transport yourself nearby a living being.
Magic items store psychic energy to perform their effects. Every magic item is psychic in some minor way, to release and control psychic energy based on a simple set of instructions. The more powerful an item is, the more complex and person-like its artificial mind is; hence ego weapons.
Finally- Magic as Psionics also explains all the ridiculous wild magic, magical crossbreeds and beasts, and dangerous or cursed magical items and places. The random thoughts and feelings of a mind could implant themselves on an item or location, even without psychic powers, and along with insane or dreaming Wizards could create unforeseen creations of pure imagination.