Why ask a question and then spend the next half-a-dozen sentences answering the question yourself. Firstly, you generalize, saying 'What makes oldest children..." Then you drop back to, 'What makes so MANY oldest children." You've already answered the second part to some extent, namely, by saying the responsibility parents so often load upon their eldest to look after and set examples for the younger ones.
But it is a generalization. My elder sister, four years older than me (I was the middle child of five kids) didn't seem to have this problem. But then, we did have a full time, stay-at-home Mum, and not someone who had to duck off to work leaving an elder child to 'take the weight.'
It could well be that the 'Elder Child' syndrome has been exacerbated by the two-parent working family. That is, being obliged to undertake the work to keep house and home together.
This trend started around the 1960s and continues to this day. Though there are a few stay-at-home Mums still around. Prior to that, there were distinct lines of demarcation between what Mum did and Dad did.
I'm not advocating we attempt to 'turn back the clock,' but simply to explain why, there is so much angst in so many elder children today.