Setting GPU rendering: CVMCompiler has now run for 5 hours I've got an AMD 6750M graphics card with 1 GB of VRAM and Blender 2.79. I successfully selected the card as a GPU target in the user preferences.

When starting my render, Blender says: Loading render kernels

According to Activity Monitor a process named CVMCompiler has now been running for over five hours. That surely is more than a few minutes?

The rest of macOS is practically unusable, as it seems most things are reliant on the GPU. When I force quit the process, other apps, including Safari, become responsive immediately.

What could be the issue? I can't see any signs of any errors, but where should I look to find any?
5 Hours loading kernel doesn't sound right, that should be the first indication that something is wrong.

Anyway your graphics card is too old and unsupported.

As stated in the Official Blender Manual only cards with GCN (Graphics Core Next) 2.0 or above are supported. 
That means Radeon HD 7790 and above, as stated in the linked Wikpedia article

The AMD 6750 is no longer supported. Also 1Gb of memory is very low for today's standards and you probably wouldn't get any benefit from GPU rendering anyway