The argument that you can reskin something is a weak argument because it holds no merit.
That's just saying the same thing twice. Reskinning has clear merit because I do it all the time to create my concept of a character from existing classes. Not just D&D.
"Anything can be reskinned" is actually the bad argument because it acknowledges the lack of need for the psionic classes. Demonstrating that point with other classes doesn't disprove that it also applies to psionic classes; it reinforces it.
Wizard do not cover the standard class. Classes that start with magic and the subclasses at 1st level work better because then it's not a case of "I think I'm a psion just give me some more experience to find out".
Bloodlines do cover that well. The force was hereditary and ran in families, after all. Same with mutations and the X-men.
They can if they want to. Neither professor X nor Obi-wan need to, though.
And yet they do it. That demonstration of manifesting power is common in the tropes. Glowing eyes and pointing gestures and more. Bleeding noses if you prefer. It's also D&D.
That argument that there wasn't such a tell or display of power didn't exist in 3e or 4e. There were descriptions on demonstrating that manifestation. If a DM handwaved it away, it wasn't what we had in the last 2 editions. It was the standard expectation. Just like the magic / psionics transparency (they are the sae and affect each other equally) was the default rule. Psionics is Different was an optional rule.
4e still followed that AEDU power structure and have very few encounter powers, enhanced at-will powers using the power points (like sorcerers do with sorcery points). 3e used a spell point equivalent variation for powers and manifested them under the same rules as spells, and recovered them under the same rules as spells. The only difference was in psionic feats that required the reserves. They also couldn't just manifest a power when grappled or bound -- they needed to make concentration checks first. Concentration was important.
The way you describe psionics hasn't existed in D&D since 2e, and that required making ability checks in order to manifest powers on top of spending power points.
It's not that I disagree with the existence in your concepts in media and pop culture. They just don't match what the default rules in D&D have been. D&D matched other tropes where tells and concentration were a big thing.
Wizards still don't cut it as the standard in any case. I see the UA subclass as a wizard specializing in psionic related magic, not the actual psionicist.