Well written article. Simple and clear.
Hegel is more sociological. Schopenheuer is more metaphysical and tries to acknowledge realities beyond the normal human experience. Though why this is dubbed as pessimism I am not clear.
It is obvious the we are not in control of the world or even our own lives. Recognizing this situation is the way forward to humans both as individuals and as a society.
To an extent you are answering your own question. Hegel would be seen as arguing that we are in control, that progress is made. Now I'll caveat that, the rest of my post, and the original article with the warning that there's a lot of simplification going on. That's one reason why the article is clear.
It works by positing the idea of Hegel and Schopenhauer being directly in conflict and as Dicky U's quote covers it's east to illustrate Schopenhauer's animus against Hegel. As I've mentioned in reply to Dicky on another thread though I think viewing the philosophy of one philosopher as a dirdct answer against another isn't correct even if the philsopher tyemself says it is but it's a valid perspective to take one view of the work.
As to Schoenhauer's pessimism, it's both in contrasr to Hegel, and based on the idea that humans are destined to strive for the unachievable. And anyone who writes 'Life is a constant process of dying' is likely to he seen as a pessimist. Add to that a life lived pretty much like that and the tag fits for most people.
In the related conversation on the Kant thread, I suggested that there's a book to be written that looks at invidual philosopher's apparent 'neurodivergence' based on their lifes and relate that to their philosophy. I suspect you could write a book on Schopenhauer alone
