For large teams, just solid sound skills as it's easier and hypothetically better to keep people focused on just their main proficiency and what they enjoy doing the most and hire people that master and are willing to work on that one area. It's not wise to overload people with work or multiple roles and there might not be need for some kind of middlemen, depending on the organization. For small teams, multidisciplinarity and the ability and willingness for changing hats, because you may not be able to have or cannot afford to have a separate specialist for every task.
Technically the common audio middleware software are developed by 3rd parties and they are so polished and usable by sound designers out of the box that I think most custom, technical audio-related work in game development is likely to be so extensive and technical that it'll require time and a team of experienced programmers that are primarily programmers. Or then the technology could be bought from another software house. Or then such technical work would be so small that it could be done by a programmer or programmers whose main responsibility is something else. Although of course it all varies depending on what the project is like, how the company operates and what kind of professionals they have there. In general, artists don't develop tools/software for their own purposes, but rather tools and software that are needed by artists are developed for them, or then the team uses common middleware, which artists should be familiar with or able to grasp it quickly. Operating custom tools is taught. Artists may in some cases and roles need to code small scripts for controlling and configuring the tools e.g. selecting the audio samples, playing back audio samples, setting parameters for effects etc.
I would guess that a "programmer-audio designer-composer" is not the primary interest of most companies in the current and the very business-oriented mainstream game industry, where most tools can be bought off the shelf and the talent pool is already so large that companies can pretty much find and pick whatever talent they need (although some sources say that there's shortage of highly skilled people who are willing to work on games / in the game industry, I would assume this to apply mainly to technical positions though), but it doesn't mean that such skillset wouldn't still be valuable in some special cases and for special lines of work. It's always useful to know about different things e.g. programming, especially when everyone mostly works with software anyways, and if you enjoy learning new things. Plus, no-one's saying that one should do like everyone else seems to do, so one might be that "programmer-audio designer-composer" and actually find work where one gets to apply that kind of multidisciplinary skills, if one wants to. In a competitive situation it might be just that multidisciplinary background that gets you the job, but it depends on what the employer is looking for, and it might as well not be all about technical and vocational skills, but more about just personal traits and how they fit for the role. If coding interests you and you enjoy it, there's nothing stopping you from writing code.