I'll answer this question in several parts:
Why do AGI systems need to have common sense?
Humans in the wild reason and communicate using common sense more than they do with strict logic, you can see this by noting that it is easier to appeal to someone's emotion than logic. So any system that seeks to replicate human cognition (as in AGI) should also replicate this tendency to use common sense.
More simply put, we'd wish that our AGI system can speak to us in common sense language simply because that is what we understand best (otherwise we wouldn't understand our friendly AGI would we?). Obtuse theory and strict logic might technically be correct, but don't appeal to our understanding.
Isn't the goal of AGI the create the most cognitively advance system? Why should the "most perfect" AGI system need to deal with such imperfections and impreciseness present in common sense?
First, it might only appear to be the case that common sense logic is "irrational". Perhaps there is a consistent mathematical way to model common sense such that all the subtleties of common sense are represented in a rigour fashion.
Second, the early study of Artificial Intelligence started in the study of cognitive science, where researchers tried to replicate "algorithms of the mind", or more precisely: decidable procedures which replicated human thought. To that extent then, the study of AI isn't to create the "most supreme cognitive agent" but to merely replicate human thought/behavior. Once we can replicate human behavior we can perhaps try to create something super-human by giving it more computational power, but that is not guaranteed.
I still don't see why common sense is needed in AGI systems. Isn't AGI about being the most intelligent and powerful computational system? Why should it care or conform towards the limits of human understanding, which requires common sense?
Perhaps then you have a bit of a misaligned understanding of what AGI entails. AGI doesn't mean unbounded computational power (physically impossible due to physical constraints on computation such as Bremermann's limit) or unbounded intelligence (perhaps physically impossible due to the prior constraint). It usually just means artificial "general intelligence", general meaning broad and common.
Considerations about unbounded agents are studied in more detail in fields such as theoretical computer science (type theory I believe), decision theory, and perhaps even set theory, where we are able to pose questions about agents with unbounded computational power. We might say that there are questions even an AGI system with unbounded power can't answer due to the Halting Problem, but only if the assumptions on those fields map onto the structure of the given AGI, which might not be true.
For a better understanding of what AGI might entail and its goals, I might recommend two books: Artificial Intelligence: The Very Idea by John Haugeland for a more pragmatic approach (as pragmatic as AI-philosophy can be, and On the Origin of Objects by Brian Cantwell Smith for a more philosophically inclined approach.
As a fun aside, the collection of Zen koan's: The Gateless Gate, includes the following passage: (quoted and edited from wikipedia)
A monk asked Zhaozhou, a Chinese Zen master, "Has a dog Buddha-nature or not?" Zhaozhou answered, "Wú"
Wú (無) translates to "none", "nonesuch", or "nothing", which can be interpreted as to avoid answering either yes or no. This enlightened individual doesn't seek to strictly answer every question, but just to respond in a way that makes sense. It doesn't really matter as to wether the dog has Buddha-nature or not (whatever Buddha-nature means), so the master defaults to absolve the question rather than resolving it.