At this point, I think everyone has heard how bad the translation of Cross Channel is, but I was wondering if a person who read it in Japanese and another who read it in English could have a coherent discussion about the *events* of the story, disregarding the nuances of writing, subtle themes, etc.

In the most technical sense possible, I guess so.
Both English translations of C+C completely disregard Romeo's incredibly precise word choice which consequently loses the subtext in his writing, which in turn defeats the purpose of many scenes and renders the vast majority of the game as pointless filler when that isn't how it comes across at all in the original. To make matters worse, they also contain hilariously bad translation errors that make many events come across differently. Even Ixrec himself acknowledges that his translation of the game is pretty bad and GHS even goes so far as to insert his own immensely retarded interpretation into it, but we can disregard all that for the sake of this particular argument.
Allow me to use this grossly exaggerated visual example to prove my point and illustrate why this is kind of a silly train of thought. Let us take a look at the two images below.
The image on the left is Leonardo Da Vinci's Mona Lisa and the image on the right is my attempt to emulate it in MS Paint. Both of the drawings depict a smiling woman with her hands crossed, right? So if one person sees the drawing on the left and one sees the drawing on the right, they can both technically say they have both seen an image of a woman smiling with her hands crossed and discuss the two on that level, but at the end of the day its just NOT the same thing and one is very obviously a poor imitation of the other with all of the "subtleties" and "nuances" removed, imo this just defeats the whole purpose.
The best part of this whole thing is you have a bunch of people running around claiming the image on the right is just fine and that you aren't missing anything by not seeing the one on the left, which is highly aggravating.
I hope this helped clear up the issue.