I think it makes sense. The stated reason to destroy the base is to destroy the cure. The Salarians are distracting an unknown but large number of enemies so Shepard can get in and plant the bomb. They aren't taking over the base and really don't have time to poke around. (Yes, I know you have infinite time, but that's just for gameplay reasons.)
I guess Wrex could follow Shepard's team grabbing up any datapads he can, but the argument is over destroying the base at all.
There's no reason Wrex couldn't just download any info they had on the cure into his omnitool while everyone else is doing all the fighting, or at least no reason given. Sure he might have been short on time but it's better than nothing. Also the arguement was over destroying the cure, not necessarily the base or stopping Saren.
I could see Wrex telling Shep to stuff it and doing a wreckless krogan thing where he just goes lone wolf trying to find that cure. Everything Wrex says leads me to think it's probably something he'd see as worth risking life and limb for so why not? What he's willing to shoot and kill good guy shep when surrounded by a dozen or so badasses (A few with plot armor) over the matter but wont make an all out assault on a compound of easily destroyed geth?
You could even make it how he dies instead of getting shot by that ****** Ashley. He goes in and is looking around and loses track of time and blows up.
Look, my point at the end of the day isn't whether or not Wrex would have actually found something useful or not. My point is that nobody even considers this as a valid option. The thought to just split up and do their own thing doesnt cross either party's mind. they both just decide there's an impasse when again, that's not strictly speaking true. Wrex looking for data on the cure within the time allowed while shep tries to blow the place up are two actions that do not conflict with each other in any way.