I'm glad to see an interesting discussion other than just a blanket argument on whether or not there should be 5e psionics.
My attempt to lay out...
Why I think that in fiction the term "Psionics" and "Magic" are generally interchangeable but then I disagree with myself and will go on to say that in Dungeons & Dragons psionics should exist and should be unrelated to Magic.
In the Forgotten Realms it is established that there is a "thing" called The Weave. It can be destroyed, it can be built, it can be altered, and for some characters it can be interacted with. A 5th edition character in 5e might interact with the Weave by casting spells, having a magic item, using a scroll, using a spell-like power, or many of probably hundreds of different items and abilities. If, in 5e Forgotten Realms something is "magic" then it works via using The Weave as its method of making changes on the plane. Arcane spells use it, powers granted by gods use it, if you think "Primal" is a source then even Nature itself uses it (midichlorians???).
In 5e there is also, clearly, the ability to make changes on the plane in a purely physical manner. You can chop down a tree or punch someone in the face and knock them out. There is an expected level to what you can accomplish (you can jump 10 feet but not 40 miles) based on a level of science that roughly equals our own world. In the REAL WORLD giants can't exist because reasons (I want to say oxygen levels in the atmosphere and inability to exchange gasses, but i'm not quoting it because its not important to the conversation) however there are plenty of giants in the Forgotten Realms. This isn't because The Weave is required for giants to live, but because the laws of science in the FR are not our real world laws.
So the question conceptually about psionics is.....can there be another force in the Forgotten Realms that allows work to be done on the plane, but that isn't The Weave or physical actions? If you believe that "Psionics is different" then you are saying yes, there is this totally different way that this can be done, and it doesn't use the weave. I frequently hear people say that the idea of disentangling something from magic (The Weave) is just a powergame move, but in actuality there is already a core class that does this "affect the world in a way that uses some other power than The Weave to let me do something that breaks physical laws of the Forgotten Realms as we know it, and that class is the Monk.
The Monk uses "Ki", a resource that is not explained in detail as to how it works to do things like attack faster, use telepathy, avoid negative effects, and stun enemies and yet not one of those powers uses "The Weave" nor can they be stopped by a counterspell or anti-magic field. The monk can even use this nonmagical (but non-mundane) "Ki" energy to affect The Weave and cast spells if they take the correct path to do so.
I like the idea of there being more to the world than just "Magic" and I like to think of Magic as being synonymous with "The Weave". I like to think that some characters have power sources other than magic, and I like to think that all the different sources CAN interact with each other but don't necessarily HAVE TO interact with each other.