Hi,
The live loader can only work on the results page – that is why it is called results page live loader.
The main reason is how different the shop/archive pages behavior is from the results page. Unfortunately if a search query is set, WordPress automatically redirects to the results page, which is usually very different from the shop/archive pages, and it is extremely difficult (or in many cases impossible) to replace the contents on those pages (which the live loader must do to work properly)
We are actively working on possible solutions, so I hope that in a future release this feature can be available, it would be really awesome for us as well.