How to manage a developer who has poor communication skills

Question

I manage a small team of developers on an application which is in the mid-point of its lifecycle, within a big firm. This unfortunately means there is commonly a 30/70 split of Programming tasks to "other technical work". This work includes:

• Working with DBA / Unix / Network / Loadbalancer teams on various tasks
• Placing and managing orders for hardware or infrastructure in different regions
• Running tests that have not yet been migrated to CI
• Analysis
• Support / Investigation

Its fair to say that the Developers would all prefer to be coding, rather than doing these more mundane tasks, so I try to hand out the fun programming jobs evenly amongst the team.

Most of the team was hired because, though they may not have the elite programming skills to write their own compiler / game engine / high-frequency trading system etc., they are good communicators who "can get stuff done", work with other teams, and somewhat navigate the complex bureaucracy here. They are good developers, but they are also good all-round technical staff.

However, one member of the team probably has above-average coding skills, but below-average communication skills. Traditionally, the previous Development Manager tended to give him the Programming tasks and not the more mundane tasks listed above. However, I don't feel that this is fair to the rest of the team, who have shown an aptitude for developing a well-rounded skillset that is commonly required in a big-business IT department.

What should I do in this situation? If I continue to give him more programming work, I know that it will be done faster (and conversely, I would expect him to complete the other work slower). But it goes against my principles, and promotes the idea that you can carve out a "comfortable niche" for yourself simply by being bad at the tasks you don't like.

I want to clarify that I'm not trying to address this issue due to a grudge, or that I have a "chip on my shoulder" as was mentioned. I'm looking for advice on how to keep a well-rounded team, which is happy and motivated. By observing the variety of answers to this question, it seems like there are a lot of different opinions on how to achieve this.