If you start by replacing menial labor, there will be more unemployment but you’re not going to build the political will to do anything because those jobs were seen as “less than” and the political class will talk about how good and efficient it is that these jobs are gone.

You need to start by automating away “good jobs” that directly affect middle/upper class people. Jobs where people have extensive training and/or a “calling” to the field. Once lawyers, software engineers, doctors, executives, etc get smacked with widespread unemployment, the political class will take UBI much more seriously.