Let's say someone broadcasted a solution to the recent block. Why, as a node, would I accept it? I can just copy and broadcast it myself instead. Yes, most of the times the first node would win, but in rare occasions I would get ahead of that node (mine would propagate faster) and get the reward, no?
Well, you could. But you would not get the reward.
The mining reward receiver is added inside the block as
miner) and therefore you can't change that without remining the block. Here's some more info: What is the beneficiary in block header?