Instead of reading the mining rate of the oil rig, I'd much prefer to read the internal oil rig storage. This would make the LTN integration better, as it would be possible to call a ship only when it has enough oil already.

I concur with this.

I did some research, and I understand why it's not the case: it seems really hard to do. Currently, the oil rig is basically a pumpjack with a modified image, and modified outputbox to make it bigger. The circuit connections are the ones the game provides for a pumpjack.

The only way I could think of making it work, was adding an invisible storage tank that takes oil, then making the circuit connection go there instead of the pumpjack. However, since I'm not really a Factorio modder, this is not something in my capabilities. I do hope somebody with the skills does it. It seems the author accepts pull requests on github.