[P8riot]

While Jupiter’s gravity could very well deflect or otherwise modify the trajectory of these Near Earth Objects, thus preventing their impact on Earth, could it not also deflect an otherwise benign object directly into our path?

[SunkenCiv]

That’s the idea. Also, there’s enough crud in the general vicinity of Earth that, over time, most of the known impacts, including major ones, could have been caused by stuff that is nearby. For a rough analogy, think about the toy from the 1960s, the Spirograph.

[Phsstpok]

While Jupiter’s gravity could very well deflect or otherwise modify the trajectory of these Near Earth Objects, thus preventing their impact on Earth, could it not also deflect an otherwise benign object directly into our path?

Clearly, but the math makes it hugely more likely that any deflection would be onto a path that doesn't intercept Earth than one that does.  After all, there are so many more paths that don't hit us than the limited set of paths that do.  But it's still possible.

I have a spread sheet that I pull up every time a new Earth orbit crossing NEO is spotted and the odds of it hitting us are bandied about.  I take our orbital radius and uses that to calculate our orbital circumference.  From there I divide it by 365 days to get how fast the Earth is moving in our orbit.  I use that to calculate how many minutes, hours, days, etc. the calculated path of the NEO needs to be off for them to be wrong about said objects hitting us or not.  Usually it's pretty big, but a few times it's been a matter of a few minutes off for a prediction that's years away, making a really tiny error have potentially big impact (pun intended).  And the guys who usually make the most certain statements about such things are the same guys who forgot to convert from from meters to feet and sent the Mars probe straight into the planet instead of into an orbit.  Ooopsy.

[bert]

Also, as it attracts objects, it grows. As it grows it’s attraction of earth increases. A growing Jupiter and a burning sun mean earth is moving away from the sun and towards Jupiter......

Result, global cooling.

[burzum]

While Jupiter’s gravity could very well deflect or otherwise modify the trajectory of these Near Earth Objects, thus preventing their impact on Earth, could it not also deflect an otherwise benign object directly into our path?

It is difficult for Jupiter to do that. It is hard for Jupiter to scatter an object into an orbit that crosses ours without also kicking it out of the Solar System.