I'm somebody who really wants Blackmagic cameras (BMPCC4K, 6K, and 6K Pro as well as the BMCC6K) to support binned shooting modes,
because it can potential [url=https://www.adimec.com/reducing-noise-and-increasing-camera-frame-rate-through-binning-on-sensor-binning-versus-digital-binning/]offer lower resolution, higher frame rate modes with less noise and larger field of view than cropped modes.[/url]. I hadn't considered using binned readout as a way to decrease power consumption when idle but I think there's some potential.
There's probably a bunch of behaviours that Blackmagic can have for special cases that would lower power consumption little by little but they probably prioritize usage of their FPGA which I assume is what accounts for most of the power usage of their cameras. For example, you could read out a 4K image from the sensor, debayer it, then scale it down to 1080p for preview. I'm assuming this is what Blackmagic does. Instead they could skip debayering by averaging the two green pixels in a Bayer quad and using the red and blue as-is. That would definitely use much less of the FPGA and less power while colorizing and downscaling the sensor data in one step. However since the screen is 1920 x 1080 it would still need to do some scaling to fit a 2048 x 1080 image on-screen without cropping and it would still need to debayer and scale when shooting with an actual 1080p crop. So it needs to use scaling for a bunch of things, including zooming for focus, and it needs to debayer for ProRes or BRAW but there would only be few scenarios where it could use the green averaging mode.
That being said, I don't think debayering or scaling consumes all that much power and I don't think they're major contributors to the difference in power usage you recorded. For example, since 1080p is 1/4th the resolution of UHD, I would expect 1080p60 should still use less power than UHD30 despite being twice the frame rate. Instead they use the same amount makes me think sensor readout is a big contributor. In that case, a binned readout should provide similar power savings.
The only downside I could really see to binning in idle is that the zooming and focus peaking which wouldn't work as well. I'd imagine everything else like frame guides, false color, zebras, sharpening, and LUTs are only happening after the image is downsampled to 1080p anyway so they should be fine. As long as the user knows this is the case though and the mode can be toggled though, it should be a fine trade-off.
This obviously all assumes that the sensor's Blackmagic is using in these cameras supposed binning on the sensor side otherwise the advantages would be negligible. I've heard people say the P4K uses the Sony Starvis IMX294CJK but that was made for security cameras so I'm sceptical. The IMX299CJK came out the same year and has the same specs so I'm under the impression that might be it. The former specifically mentions a binned readout mode that can do 120 fps 2K using the whole sensor but the latter just says to check the datasheet for readout modes and I can't find any mention of binning mode for it. Same is true of the IMX410 which is apparently used in the BMCC6K.