I am interested in trying some HDMI-output devices, but having that output piped into my computer. Is there something like an "EasyCap" for HDMI, piping the data in through perhaps something like USB3?
In order to pipe video into your computer, you will need a video capture card. There are indeed USB devices that can handle this, but they are not of the 10-cent USB adapter variety.
Here is an example of the sort of hardware that is designed for the task that you want to accomplish, but the price is most certainly not in the 10-cent range that you were likely hoping for: