I have heard of people dragging an old main sheet or similar behind the boat to grab in this situation. You would have a small reaction time to swim to the line as it goes by you. I think if I was sailing alone in good breeze this is what I would do.
I seriously doubt you would ever be successful at grabbing the rope AND holding onto it. For starters, if your line is 50 feet long and the boat is travelling at 15 MPH, that gives you just over 2 seconds to orient yourself, find the line, swim over to it and grab it. Keep in mind that the end of the line is moving at 15mph, is virtually invisible, and could very well be underwater. Then, once you grab the line, you have to try to hand hold onto a boat which weighs 350lbs or more and is travelling at 15MPH or so. All I can say is good luck.
Unless you're sailing in an offshore race like the Tybee 500 where you're miles away from land and other boats/sailors, there is really no need for a tether or drogue. If you follow the age old practices of 1) notifying someone on land of your expected departure and arrival times, 2) sailing with other people whenever possible, 3) sailing within your ability, 4) staying within sight of land, and 5) always wearing a life jacket, then you should be safe. It also dosen't hurt to carry a cell phone or VHF radio.