Hmm, I'm not sure about the link to lower birth rates. Birth rates have been falling in e.g. Western and Northern Europe for a long time, despite strong social safety nets.
While I think the link between birth rates declining and automation does make sense, it will take quite sometime for this to verifiable as this is a somewhat recent anxiety. The reason for the trend that we have seem over the last decades seem to mostly stem from lower childhood mortality rates, women having access to the job market, and perhaps to a lesser extent climate anxiety.
Young people can’t afford life bringing kids isn’t a smart idea. So the link is definitely there.