As someone who's spent a good deal of time in the Caribbean there's some massive misconceptions here.
1. The Bahamas are so close to Miami that you can see Bimini from the time you take off from the Miami airport. Literally, closer to go almost anywhere in the Bahamas from Miami than to Jacksonville from Miami. Puerto Rico and the Virgin Islands and Miami are all 1000 miles away from Miami, a 3 hour flight. It makes as much sense to compare living there to the Bahamas in a discussion about being close to Miami as it would to suggest living in Boston or Oklahoma City!
2. The Bahamas are a series of islands. Freeport and Nassau happen to be crime ridden and not very nice. They are 2 out of the 30 inhabited islands in the Bahamas and not at all representative of the rest of the country. If you've formed your opinion of the entire Bahamas based on an experience in either of those two places, you're doing the equivalent of assuming Seneca Falls, NY is the same as a bad neighborhood in the Bronx, because they're both in NY right?
3. Statistically you're far safer from hurricanes in the Bahamas than in wide swaths of the gulf coast of the U.S. See point 1 above, Miami is at the same elevation or lower (there are actually some small hills in much of the out islands, not so much South Florida) and essentially in the same spot as the Bahamas. And has already been pointed out, there are inventions like generators and solar panels that can provide power absent the grid and it takes no time to restore internet after a storm if it goes down at all.
All to point out that the OP asked why not the Bahamas vice Miami, so riffs on Puerto Rico and hurricanes and crime don't make a lot of sense in that context.