The right too for the job most certainly is not HDMI.
The video feed should (depending on usecase, sure) be compressed on the device and sent over USB.
Sending uncompressed video just to be badly compressed in a capture device is most definitely not the right tool for the job.
The camera already does realtime compression to the sdcard. It has dedicated hardware for this. USB-2 has adequate bandwidth for compressed audio+video.
Your HDMI capture device (which is a cheap portable device with limited processing power) is probably going to do a much worse job.
Sending uncompressed video over usb is absurd.
On these cameras, HDMI is the right tool for the job. USB video quality is often poor where it's supported, and HDMI is there for video output.
These cameras are not made to be webcams. OP is using theirs as one, and that's fine; I do too. But device-side compression for USB video out, a webcam app, etc. are webcam features. They come at a cost, and many camera buyers don't need them.
For those of us using these cameras in these nonstandard ways, we can reach for HDMI, which is the right tool for this particular job.
It's a workaround for the camera not bundling all the features that it needs to be a webcam, absolutely.
The standalone cameras I've used haven't included free webcam functionality and I don't think that's outrageous, but apparently many people here who've been downvoting me disagree.
Personally, I think HDMI is great for A/V tasks that a camera doesn't support out of the box since it's a widely supported standard.
Realtime compression in a portable device with limited processing power is going to reduce quality. It is better to transport uncompressed video and let the receiver decide how to manage it. USB-3 has adequate bandwidth for doing this. USB-C lets you switch to DisplayPort if the receiver can handle it.