I guess that could work but if you only have a bit of prep time you can't have any blocks and stock cases will probably be the only viable options. In addition I feel like a unified topic allows for a variety of perspectives on each and ability to prep for various opponents based on the cases that they're running
I think we have different ideas for what a "block" is in the context of a debate (my experience with a block is two speeches from the same side back-to-back, which I'm guessing is not what you're referencing here), so I can't speak to that. A lot of people do run stock cases, though there are easy work-arounds that prevent that kind of behavior (pick debate topics that aren't absurdly common and aren't too vague) and I've only rarely struggled to come up with a case on the fly.
I'm just generally not as big of a fan of a unified topic. That's almost certainly a result of my experiences, but I've generally found that unified topic debates can get pretty repetitive between rounds. That's not to say that there's no room for creativity and variety, but I've definitely had tournaments where there was great creativity and ones where there wasn't so much. You also run the risk of that single topic being Pro- or Con-slanted, which is, of course, a concern if you have multiple topics as well, but isn't as all-encompassing of a problem in that instance.