Of course it's not a requirement. There are many, many startups that do great without any social media presence, but they are less visible (duh!) and so it's easy to forget they exist.
Bear in mind that a lot of the startup wisdom you see out there is regurgitated content marketing.
Take for instance startups that create products for other startups: payroll, customer support, developer tools, lead generation, mailing list software, and so on. These startups write about the lessons they've learned and they share all their best tips and tricks, but their goal is to build an audience to hawk their products to. The less observant consumers of this content take it at face value and believe these blogposts represent some deep startup wisdom when in reality they're reading advertorials.
You need to generate sales leads somehow. Depending on your product social media or content marketing may be a good way to accomplish this. But it's not required and in most cases your time is much better spent elsewhere. It takes a long time (2 years easily) and a lot of effort to build a social media audience, and even then you may not end up with an audience that is really interested in buying your products.