The heaving line in a bag is designed to be thrown... not tied to a life buoy, although it could be used that way.
The reason for not securing it to a rail is to allow you to move around the vessel to get to an optimal place to heave it. To avoid having someone forget to hold on I tied a loop in the bitter end, which easily slips over your wrist or other fixed object.
We have used our heaving line several times docking
or rafting and found it useful to get a line to someone on the other end.
I agree with DAVID M about the strobe light... Our lifering has a commercial
grade strobe life with a 6 foot polypro line spliced to both of them. It would be deployed immediately after some one went into the water to mark there position and give you a reference to sail back to.
I used to crew on a ship in San Francisco
Bay (USS Potomac) and performed countless MOB drills with a lifering and strobe. Both in daylight and dark, the strobe light was a good reference to steer too. Off shore a strobe light will be a beacon, even with large waves.
We also carry a lifesling, which has a polypro line to drag it through the water and hopefully get it to the MOB once we have located him, if needed.