But I think that doesn't mesh well with what a lot of people who identify as designers see as their role. From your portfolio you clearly don't fit the stereotype I'm describing, but a lot of self-identified designers (especially those with an agency background) want a lot of power to control others, rather than direct ability to create themselves. That is, they define "design" as something separate from coding, shipping, supporting, and studying the usage of a product.
That kind of hands-off approach to design won't work in a startup smaller that a dozen people or so; there just isn't enough of it to do. And even if there were, that sort of designer becomes a bottleneck, slowing iteration and reducing effectiveness. That's especially bad at small startups, because, as you say, at least one founder already believes they can do a lot of that work.
In your shoes, I'd look for a company where everybody is involved in product research, strategy, and prioritization. Sure, you'll do a lot of front end implementation to begin with, because that's what a startup needs. But if you also pick up some product tasks (e.g., organizing user tests, distilling site data into useful feature guidance, solving business problems via feature changes) I think you'll end up with the kind of involvement you want.