The satellite antennas used to uplink a video feed like this have a very small beamwidth (around 1° depending on antenna size and the frequency they're using), so you need to keep them pointed precisely at a specific point in the sky. They rely on a sensitive IMU (accelerometers) and fast motors to actively track the satellite while the platform moves. They're quite impressive to watch in action[1], but I'm guessing they're no match for the shock and vibe of the rocket landing.
They use a satellite link on the barge, and the vibrations and pressure from the landing booster almost always cause the link to drop until things calm down.
They have said in the past that it's "solveable", but it's not exactly a high priority by any means as they can always retrieve the footage when they get to the barge.
According to the SpaceX media commentary folks that were talking as it was happening, it's because "intense vibrations are interfering with the antenna".
They both say Oh exactly at the moment when the smoke clears, meaning that they themselves are looking at the live feed while pretending it was lost.
Also the chances that Spacex doesn't have any other presence next to the droneship are zero. There in no reason for them to be beaming the data feed via a finicky satellite connection only.
Besides, I am sure the rocket itself had radios that were reporting on what was going on. When they all stopped sending signals at the very point the rocket should be touching down, that was a pretty sure signal it had crashed.
Not a manned boat, but it seems obvious that they would want to have another unmanned boat with instruments recording the landing event. This could be much smaller and thousands of meters away from the landing site so there would be very little risk of it getting inoperable by the landing rocket.
They keep people in the area where rockets liftoff and land (a few kms away) and besides, they launch rockets in space, do you really think they lack the technology to, for example, send the video feed to another ship waiting at a few kms distance?
