How can I have device-specific block layouts?
I followed the counsel of Berdir and Ash Upadhayay. Having a sub-domain dedicated to mobile devices is a sound advice as many important websites currently use this method. As they say, using the same domain for both version could cause problems with caching and referencing by indexation robots.

This however doesn't clash with my requirements, and I was able to fulfill them using the Drupal 8 version of the Domain Access module.

The Domain Access module allows to define specific sub-domains that Drupal 8 will be able to recognize. The module then makes possible to condition all manner of things depending on the current domain, including the block layout. 

I installed the module itself and its contrib module (packaged with it) Domain Alias. This last module allows you to associate various patterns to a domain that works as an alternative way of detecting the domain. This allows to match the URLs of local installs of the project without changing the Domain configuration.

This is the perfect solution for me as I am now able to condition the display of blocks from a single page - the block layout page.

All that's left to do then is to redirect the user on the correct sub-domain depending on the device, using a module such as Mobile JS Redirect.