Time, money, staff...money.
SpaceX has prided itself on being a lean company, with very little of the bloat that affects NASA as a government agency.
You want floating FO lines between the barge and either the support ship or a second barge.
Who is setting that up? Who is testing it? Who is reeling it all in when done?
The resources you would devote to that SpaceX is instead devoting to other areas that have been deemed more important.
That's it. Why bother providing the feed in the first place then? Because the system they DO have is cheap, easy to set up, and doesn't require a dedicated team of people to maintain and fart around with. Why offer it at all then? Because it's cheap, easy to set up, and doesn't require a dedicated team of people to maintain and fart around with.
This is all cost vs. reward. The cost for them in the current system in time, material and labour is minimal. The reward would be a live video of their 1st stage returning successfully in a world first...where they would probably provide an on-site HQ recording at a later time anyway.
In your system the cost is WAY higher across the board. The reward is letting people watch repeated failures as they iron out the bugs and try to figure out an immensely difficult engineering problem, or, on the off chance it works fantastically and people get to see that world first live....they provide a recording in HD the next day taken from the onboard camera system.
You've said yourself they are very good at what they do. Don't you think that they COULD figure out the live feed issue on the barge if they wanted to?
This is a "good enough" system as far as SpaceX is concerned. Otherwise you can be damned sure they WOULD have fixed it long ago.