You don't even need to use HDMI or have a capture card; you can connect the camera with USB and use gphoto2/v4l2loopback to present it to the OS as a video input. With older cameras that have USB2, you will be limited to a lower bandwidth and resolution (200Mbps -> 1024x680), but it still looks fantastic, especially with high-quality lenses and shallow depth of field that enables video codecs to encode the out-of-focus-areas efficiently and concentrate on detail in the focal plane.

I experimented with the old 550D of my dad. I got a successful clean picture using either USB (canon supplies a "USB Webcam Software", though the 550D is not officially supported it works. I also managed to get clean HDMI out using "magic lantern" (a firmware modification for some canon DSLRs which enables a lot of hidden features, allows clean HDMI, and on a beta branch allows "simulated focus bubtton press" to keep the camera from shutting down due to inactivity)

Also overheating. I don’t know about the Canon but my Sony had to have a number of settings changed to keep the temp down, and even then on a hot day it’ll shut itself off.

You don't need to do either. Canon offers a webcam utility driver for free, which offers a clean interface via microusb->usb.

I don't know about Canon, but the Sony one is lower resolution than the HDMI capture. I was able to buy a cheap Chinese USB-HDMI capture card for like $8 and it works actually quite well, and is driverless.