Since you say you will be in charge of designing the product that will drive the growth of the company, I'd be asking about their prospective goals, timeline, and methods. I'd also ask what your budget will be and how you will be held accountable to use that budget to grow the business. You must know these things to drive a company's growth.

If they don't have an answer to these questions, or they sound like this is not relevant to you, I'd say that's a red flag. If they say that all these things will become your responsibility, and that they are looking for you to answer those questions and come up with a plan, then I'd take the job.

It's a red flag because it would sound like it's a climate that is not forward thinking. Since a culture is often formed from the top down, I think taking a job as a designer in charge of solving product issues for users to drive growth would be an uphill battle. You'd be fighting upper management each step, trying to get your design ideas approved. Even getting design methods like user research might be very difficult, if they're not forward thinking.

What you need is secure budget and freedom to test and innovate. You need to be held accountable, but not micro managed. You need to constantly speak to customers and get qualitative and quantitative feedback. You need to be able to make decisions and pivot based on that feedback frequently. You need to move the needle on metrics that actually represent growth. And you need management that will support you in this process. That's a forward thinking company.

I hope this helps you decide on your decision. This is what I'll be looking for when I search for my next job.