The distinction does matter – but not quite in the way you were indicating. Class is not something that exists on its own, rooted in genetics say, or chemistry. Classes arise due to the economic structures people engage with to produce, consumer, allocate. If you just focus on classes, then once you have said workers and capitalists – and now you don’t look at the underlying institutions but just use those concepts – you will never notice that, whoops, there can be, and indeed is, another key class – the coordinator class.
Institutions, and the interaction with people, are at the root of it. That you are certainly quite right about. Class emerges. But then classes – groups that have circumstances such that the members have similar interests and often even similar values and ideas, and at times most importantly, shared actions and agendas, are critically important. And we don’t have to return every time we talk about and think about social relations and dynamics, to the causes of classes, etc. Sometimes we take them as a given, many of whose features we understand, and reason in light of those. Just like we do, other times, with underlying institutions like, say, markets or corporate divisions of labor, and so on. But thinking that because classes are results means we don’t need a concept for each, and we don’t need to assemble insights about each and their relations – that doesn’t make sense. It would be a bit like saying, because a cell phone, or anything else (like, say, a kidney) is a result, and dependent on dynamics of a deeper sort (electromagnetic theory, bio chemistry, etc.) we don’t need to have a label for cell phone or kidney, and to amass knowledge about them so we can talk about and think about their properties, without having to return to the deeper level. And, there is another element – some of their properties don’t owe to the deeper level, or could not be discerned at that level, because they arise from the cell phone or kidney in particular context which give them additional attributes, and ditto for classes.
Again, we opt to label something a concept, with a name, to highlight it as something to pay attention to, and to amass an understanding of its features and relations to other things, and be able to talk about it, etc. For purposes of trying to change the world, class is something to highlight – whereas red headed person, “redson,” is not…