That depends on what you mean by "small startup". In a VC-backed Silicon Valley startup where you're hiring fast and aiming for massive growth in as short a time as possible? You're probably right. Small startup that's self-funded or close to it with a tiny team, like what happens in most of the rest of the world? Yeah, you're probably rolling up your sleeves and getting your hands dirty for quite a while to come, because the code that you're selling has to come from somewhere.
Different startups have different constraints. It sounds like you're used to the SV-style startup where the constraint is time, rather than money. A lot of businesses have their most senior technical staff write code because they don't have the cash to hire more developers but their product still needs to be built.