The basic problem here is that while 'New Left' refers to a particular political tradition with a fairly clear meaning and definition, 'Old Left' really doesn't except in a sense so broad as to be completely useless. You're basically lumping together all shades of Social Democrat and Communist together as one discrete category and while, yes, they share some ancestry, so do whales and dogs.
You're right, of course, but these days I might actually play the devil's advocate and argue that the commonalities between old-school Social Democrats and Communists are actually
underrated, and that they do share a lot in common that distinguishes them from newer currents of leftist thought. There is the common strategic focus on controlling the state apparatus, as I mentioned, and a lot that goes with it such as the organization into parties (with, whenever possible, organic ties with labor). There's the Marxist lineage, as you mention, which even with all the revisions and innovations and reneging, remained clearly present in both. And there is, and I hate to be reductive here but that's honestly the only way I know how to phrase this, a certain
seriousness about political power and what it takes to seize it and wield it effectively that is completely lacking among "new left" types.