That "mature startup" I was working at had no Scrum and 6 people, including the two founders.
The lack of some sort of formal structure very much was a problem.
Granted, some people are so self-disciplined and enlightened/experienced that they are fully capable of self-organizing and don't need the support of any formal work structure. But those are rare like hen teeth. It doesn't have to be Scrum but Scrum works, so why reinvent the wheel?
This is especially true at early stage startups when there are millions of things to take care of besides the coding. The structure/being organized is super important or things start falling through the cracks.
If it is a feature request that didn't make it that is annoying but likely no big deal. But if it is e.g. a tax declaration for the company or maybe some government paperwork that had to be done, or following up a major investment/sales lead - that could literally destroy your company overnight.
So having a structure in place to make sure everyone is constantly on the same page, everyone knows what needs to be done - and mainly who is going to do it - is super important.
The problem is that most people I have met didn't realize this. Esp. when the founders have never worked in a structured team themselves, never saw how to manage a team or had a mentor to show them the ropes. So you end up with poorly reinvented management "wheels" that don't work and the company is wasting precious capital because suddenly the founder becomes an over-stressed and overloaded bottleneck trying to juggle coding, finances, sales, filing taxes and what not. Or, worse, makes the company fail because something that had to be done wasn't.
This isn't stuff that is taught in universities or that people are somehow born with. One has to learn it with experience - and ideally see how to do it from someone more experienced.
Obviously, if your entire company is one person, you don't need Scrum. However, the moment you hire your first developer you better put some structure in place because sooner or later you will outgrow the informal "system". Only so much can be done with a single person.
And once you have more than 1 developer (or any team member - it can be graphic designer, web designer, sales, whatever) you will have problems if you keep things totally free form. It becomes super inefficient.