Does it make sense to look into analytics and get more data?

That sounds like a great idea. One way to do is start small and go with a mindset to find you can tie back to realized impact. Once you do that, build on it more and more, show what/how you did, invite other people to come and contribute and go from there. Overall, it is a no-brainer; only caution balance time you put on it, don't spend too much time and don't do any feature work - it comes down to balancing short-term and long-term things.

You can also ask questions around why already you folks are not doing it or what your managers thinks of it etc.

Working at Course Hero and now Taro, it's really easy to feel the constant pressure of feature requests at a small company, but it's important to take some time here and there to think as well and try to understand the product at scale. Once you get past a few hundred users, it gets really difficult to gauge how good your product is with purely manual means, so I'm sure having some analytics would make sense in your scenario - There are very few software companies that shouldn't seriously invest into data analysis at all.

For more tactical advice on how to do this, I highly recommend this other Q&A on how to become an analytics lead.

For almost every company, it absolutely makes sense to spend time on analytics. I also like that you ask this question, since what you don't want to be is a "commodity coder" where you simply output some code according to a spec.

You instead want a job with more creativity and critical thinking. In order to enable that, you need to have broader context of what is important, beyond just the code. Data analysis gives you that!