The oil rig doesn't store a lot of oil, so for it to work, it would require to have a ship with it.
you can currently uses landfill to make an area large enough to put tanks on it, but that just looks ugly.

Maybe add a liquid tank that can be placed on the water (preferably to work with other big mods (like Angel's) that provide liquid/gas storage tanks).

It also requires an electric pole because the oil rig's power providing range is not enough to reach further if there are tanks between it and the pumps.
So either the pumps shouldnt require power (which they practically dont at the moment) or the power reach must be extended in some way.

Image of oil rigs with tanks currently:
https://i.imgur.com/wme5PpP.png

Well the current state kinda was a design choice (next to being easier to implement). Real oil rigs usually do not have a large storage, but pump oil directly into docked ships. To compensate for the lack of storage, the in game oil rigs produce oil at a much higher rate than the land based oil pumps. Personally i feel, that the current way of handling this, while different from the vanilla game, is rather balanced...

Hmm...

Although I do not work at an oil platform, I would suggest to make a buffer storage that is at least the size of one ship, so that a ship could be sent to platforms which are almost full without the ships waiting long times at the platforms.

I would suggest this in real life as well, but I assume I am missing a critical point here that makes it hard (if not impossible) or dangerous.

I guess making more ships or just using the landfill will be the answer then.