Not if you want the highest quality, and they could absolutely stop even that if desired. The only reason why those methods work is due to legacy support. If they only supported the latest versions of HDMI and DRM, it would be very hard to get decent quality video/audio. As it is, even with things currently as they are, we still don't have the high quality feeds that are sent to TVs and dedicated hardware.
I wonder if it would be possible to use e.g. an FPGA to intercept the "last-leg" MIPI signals going between a TV/monitor's control board and the physical display panel itself. Surely there can't be any DRM at that level, because there is not much more "compute" down the line?
Granted, you would have to deal with whatever your display does to the raw video signal - preferable to pointing a camcorder at the display but a little worse than the original file.
Yes actually you can. But you don't even need to go that far, there are HDCP converters that do it for you and convert to HDCP 1 (whose master keys have been made available) or just plain HDMI. See HDFury etc.