The real challenge around “responsible metrics” is only partly about the metrics themselves; it is also, and perhaps more, about how people use them.
First, I must repeat a mantra that I have used widely: we mean indicators, not metrics. A metric is something like a citation count; it doesn’t tell you a great deal, and it usually depends on the data source. An indicator is what you get when you try to assess something else, like research performance. You can’t measure research performance directly, because that requires expert and experienced interpretation, but you can create an indicator of something associated with performance, such as the average number of citations per paper after accounting for publication date and research field.