New designs shouldn't use the P-curves, because it's too easy to implement them vulnerably (for all intents and purposes any random string is a workable Curve25519 point, and that's not the case for the P-curves --- you have to do fussy input validation). But that has nothing to do with conspiracy theories about how the curve was generated.
You don't have to take my word for this; you can just read the Koblitz and Menezes paper, which takes this question on in detail.
So, no, really, none of this is true.
Yes, curves in the Web PKI are largely P curves right now, but ed25519 is also standardized.
You're certainly free to trust the NIST curves, but my point still stands: there is no good reason not to pick such constants using a nothing up my sleeve algorithm, and the fact they didn't do that means they either know something we don't, they backdoored it, they wanted people to suspect they backdoored it, or they're idiots and decided to ignore established best practice for no reason. It's entirely possible the answer is the latter, of course :)
More info: https://safecurves.cr.yp.to/rigid.html
Ed25519 is a replacement for RSA in the x.509 WebPKI, which is what I was trying to refer to when I said TLS. Classic TLS (as in non-PFS, the one that also used RSA to encrypt the session secret) is dead and nobody cares about replacing it with anything. There is no public key encryption involved in modern TLS; instead all you need is a signature scheme (for the certificate and for the final server to authenticate itself) and a key exchange scheme. The former can be ed25519. The latter can be curve25519 (specifically, the retroactively named X25519 ECDH key exchange).
My point is precisely that there's no inherent distrust in RSA (and some concerns with NIST EC, both the magic numbers and secure implementation difficulty), which is why we haven't abandoned it yet. There is certainly no inherent issue with prime field cryptography.