I can't really provide any evidence of accuracy change for you, but I do have some thoughts on free float or not with the factory handguard and barrel nut vs the Tacticool22 free float barrel nut adapter and an aftermarket rail.
For the factory handguard, it depends more on whether you're using iron sights or an optic. The rail is going to flex a little bit with or without the end cap, and you'll likely see it more with iron sights since the front sight will move in relation to the barrel. Without the endcap the rail could actually flex a bit more, depending on the amount of pressure you put on the rail itself.
With an aftermarket barrel nut and rail, the rail shouldn't flex so much (or at all in relation to the barrel), so I'd expect less variation with iron sights. I'd also expect less overall variation with the aftermarket barrel nut and rail since there is actually one less piece making up the barrel to receiver installation.
With the factory handguard and barrel nut you are sandwiching the hand guard between the chamber end of the barrel, the upper receiver, and the barrel nut. Using the aftermarket barrel nut you are sandwiching only the receiver between the chamber end of the barrel and the barrel nut, then attaching the rail to the barrel nut. It would seem that less parts in play in mounting the barrel would lead to less variation, but I have no evidence one way or the other.
The bigger factor is likely the plastic upper receiver. It seems like a metal upper receiver and a free float barrel nut would give the most solid and stable platform, and more consistent shots.
