Only had a quick scan of all posts so not sure mentioned yet, but don't overlook something really important here
Your TV settings
As a mainly PC gamer used to tweaking and faffing rather than playing

I spent a good half day on GTA5 launch day last week fannying around with everything, as I too wasn't happy with the fps and general stuttery feel (xbox 360 version by the way).
Modern TVs play a fair part in your console 'smoothness' experience these days, particularly any
'motion' settings/ options you may have.
My 360 is hooked up to a Samusing 46" LED smart TV that has all sorts of motion settings in the picture setup menus. If I have 'smooth motion' activated GTA5 is incredibly smooth, feels like 60fps+, seriously, but only for about 5 or 10 seconds then it will do a massive hitch or series of stutters. Really disappointing as it has so much potential.
I found I have to set my TV into specific
Game Mode in a sub menu to get GTA5 to play half decent. This mode essentially disables all motion options and a few others, and is supposed to be better for input lag (totally different to what I think you are getting, ie stuttering, hitching etc?)
With my TV in game mode, to my practised frames per second obsessed PC eye, it feels like GTA 5 runs at 30fps max which feels smooth when it does, but dips to around 20 or so at times, and has the odd stutter or hitch, especially when driving. But then many pc games exhibit this type of stutter, independent of frame rate, too.
I also have the play disk installed on a fast USB stick, but I honestly don't think this makes any real difference to fps or any minor stutters I still get. I get zero texture issues or pop up with this now though, and my old Elite disk drive isn't buzzing away which is a bonus.