It doesn't have to be like that, even now. When I developed the EC225 TR course there was a whole 2 hr FAM session per pilot (so 4 hrs total) early in the course specifically going through all the automation behaviour step by step. And that automation is subsequently the "normal" way of driving the helicopter around in the subsequent FAMs, especially the IF stuff but also during those (unnecessarily long, as you say) OEI procedures and much of the rest of it. In many ways its the manual bit that is the afterthought!
This was because I built the course around what I thought was needed, rather than specifically aimed at content of the LST.
So, despite the lack of "lead" from the authority in this area, it is possible for an operator to have an automation-focussed course.