You wouldn't describe a person or a family as 'BAME', you'd use the particular description of their ethnicity, if you had reason to describe it at all. BAME is more of a general term used at the population level, which I think is the reason it seems to have fallen out of favour, it's much too general to the point of being a bit clumsy.
It's a process of continuous improvement, isn't it, along with a sort of fractal approach to identity. Every time you identify a population and make accommodations to suit them, you can then zoom in on them and discover further subdivisions which you can then work towards accommodating even more specifically, should that be desired. Iterate and improve. Hopefully peoples expectations are increasing as well, maybe a few years ago the situation was such that use of a certain type of language was the best that could be hoped for, and maybe it's to be expected that we've made some progress since then.
Regarding what minorities have decided they would like to be called, firstly that lumps all minorities together, and also lumps the minorities of previous generations in with those of today. These are different people, not just between minority groups but also across generations and they are going to have different attitudes. To put them all together in one box labelled 'minorities' is itself a bit racist and is exactly the sort of thing that this improvement in language is trying to move away from.