Could I not build a script to listen to blockchain and award winners who are correct within 60 seconds of their guess?
You could, but you'd have two significant problems.
1. If you used the time (according to your own clock) that you first hear about the block from a peer, then players in your game will need to trust you to report that time accurately. You could easily adjust the time by a minute or two if it helps you or your friends win. Therefore most players will suspect that you are mis-reporting the time (especially when that time doesn't match the timestamp in the block, which will be most of the time).
2. If you use the timestamp in the block, then miners and mining pools will have an incentive to play your game and adjust the timestamps in their blocks to match exactly what their guess is. That way, they guarantee that they will win every time and will increase their mining revenue. Why would ANY non-miner play in this game, since they are really just making a donation to the miner that solves the block? Any miner that solves less blocks than another miner won't play either, since they are just contributing a percentage of their revenue to the miner that solves the most blocks. The only miner that will play will be the miner that solves the most blocks, and he'll only play if you are paying more in winnings than he is betting.