It's hard to tell really. If the barge was bigger it basically means more leanient constraints for the final approach, which could result in a softer landing. Remember that Falcons can't hover and thus have to achieve perfect timing for the final burn. If you constrain time down to almost zero and space down to a few meters and the initial velocity is terminal velocity + pretty fast horicontal movement, it's almost a wonder that they're already doing what they're doing.

The causes of the earlier failures were insufficient hydraulic fluid, a sticky valve, and a landing leg that didn't lock. I don't see any of those being helped by a bigger target.

Last time it effectively landed correctly in pretty rough conditions. The only problem was a landing leg did not lock because of ice and the stage was lost.

You're probably right. If they needed that stuff they'd probably build it. But I bet that what I suggest would result in being able to land in rougher weather with better media from 1st stage landings.

I imagine so. I'm not sure how much they care about the satellite feed, but handling rough weather would definitely be a plus. There was already one landing attempt that they had to give up on because weather was good at the launch site but bad out at sea.