Is the “Posts page” really required? Can problems occur if I leave it blank?

Technically you don’t need to include a posts page on your site. Plenty of sites consist of just some pages and no posts at all.

However, Google uses the posts pages to find links to your individual posts. If there is no such page you will need a different structure to guide Google (for instance through the menu, by including a dropdown of your child categories). Given that Google’s algorithms are secret I cannot know whether using an uncommon navigation structure is better or worse for seo.