I can imagine such a scenario as well because we are humans and we would do dumb shit like that.

FTL can mean different things. It could be the Trekkian warping space around you. It could be the Galactican jump to somewhere else. Or a portal. Or something else. Which tech it will be will matter. And different to most sci-fi, there could be more than one way if indeed c can be smashed.

The problem with space is that it isn't completely empty and stuff in it moves like evil landmines. You don't want to warp into a moon or jump inside an asteroid belt. Which means you will need this sort of highway or scenic spot system of space that is actually confirmed empty so travel along the route doesn't blow you up or jumping doesn't mean you accidentally end up in a black hole. In my mind, that would prohibit the use of the technology inside a star system because there are two many moving parts to consider and keep track of. So you need to take a year-long sub-LTS access ramp to reach the highway at the edge of the solar system before you can move safely faster than a flash. The same is true for the resource rich destination.

That's why I can also imagine a scenario where having the technology might not be as impactful as we fear and thus not lead to war. The infrastructure would have to be so massive to make this work, it isn't an immediate advantage to have it. More of a burden really, provided the economy still roughly works as it does today. And if we have the tech to reach the edge of our solar system in a shuttle bus service kind of way, we will be already enlarging our resource pool with stuff we find inside our solar system in an early Expansian kind of way. Better ROI on that than stuff from further away.