Any physical barrier to read an HDMI input?
Yep, I learnt about it but I wonder why you have to get through such cards instead of just having an appropriate software on your computer...

USB is a complex bidirectional protocol implementing a bursting bus with host arbitration. HDMI is a complex, multi-lane unidirectional protocol.
They electrically aren't  related to each other.
You need special hardware, including buffers, and packetization, to make that work.
Software is exactly that: instructions you run on a processor of sorts one data that has already been acquired by specific hardware. USB is not specific hardware for HDMI signals. They simply aren't compatible in any way.

with potentially a HDMI to USB adapter.

so, yeah, you need one of these commercially available devices.