My opinions, having over-thought this shit for almost 20 years:
Cant Neo just bend those weapons? Isnt it that the spoon isnt bending, its you, making it so he can do all these things.
My interpretation of the whole trilogy is that even though Neo became The One at the end of the first film, and could do things like stop bullets, fly, and "destroy" Agents, he wasn't at the height of his powers yet.
The One that Morpheus talks about seems like he could change literally everything in The Matrix on the code level, and Neo in the films doesn't come close to that. I wish Reloaded/Revolutions leaned more into the idea Neo is still on a journey instead of "Nope, he's the One now, no more powers or growth."
The other thing to consider is that the story of The One that Morpheus believes... might not be true. It's possible The Oracle lied to him to make the plots of the movies happen (and thus achieving true peace.) Maybe The One isn't a god in the Matrix, but Human++? The Oracle is well aware of the five previous "Ones" by the time Morpheus comes around, and all five of them represented another system of control.
Ofcourse after he assimilated Agent Smith at the end of 1, I cant even be sure what extra powers he got, but I know he was able to see The Matrix for what it was, just lines of code.
I don't think blasting Smith conferred any new power, but that's just my opinion and we probably don't know for sure.
If he sees The Matrix as lines of code, couldnt he rewrite or impact them in some way, making it so they could never harm him with those weapons?
Seeing code is different from being able to change it. I could go look up the Linux source code right now but ask me to do something specific and I'll likely have no clue without lots of research (or training, in Neo's case.) It would have been cool to maybe have a scene of him in Reloaded trying to mess with the code of the Matrix and it backfiring ("Oh shit, I just gave Deja Vu Cat an extra tail!")