The problem, in my understanding, is that the drone ship currently passes the video data to shore via a link to a satellite in geostationary orbit. Because the drone ship's dish has to be aimed within a very tight arc in order to communicate with the satellite, the vibrations from the first stage's landing burn causes the connection to be lost.

Because there'll be a ton of Starlink satellites up there once the constellation has been completed (and they'll be much closer than geostationary), the acceptable arc for the dish to be aimed should become wider, and it will likely become possible for SpaceX to use the Starlink network instead and maintain a good connection the whole time.

As I understand it, both the intermittent telemetry video from the first stage, and drone ships is the challenge of maintaining line of sight from the transmitter to an effective relay.

The antenna for reaching the starlink constellation will look more like the ones you see on the iridium handsets rather than the directional dish that is characteristic of geo-synchronous satellites. Regardless of orientation or vibration, the signals can be effectively transmitted (and received) from ground to orbit.

Now they still have the option of cutting off our access to the video when things aren't going in a way that is favorable to SpaceX of course :-).