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?