This is not necessarily specified in the documentation, but to launch Alacritty with the command to open its default terminal, it is necessary to define it as an alternative available for ‘terminal’ type applications:

sudo update-alternatives --install /usr/bin/x-terminal-emulator x-terminal-emulator /usr/bin/alacritty 50

(The number ‘50’ at the end represents the priority of the entry)

It will then be possible to define it as the default terminal, thanks to the command:

sudo update-alternatives --config x-terminal-emulator

Finally, if you want to remove alacritty from the available alternatives:

sudo update-alternatives --remove "x-terminal-emulator" "/usr/bin/alacritty"