An extremely simple argument for this sense is to consider the following two questions:
Q1. Does the epistle of Romans assume and rely on an essentially literal* interpretation of the fall of man (cf. Genesis 1-3)?
Q2. How foundational is the epistle of Romans to a Christian understanding of the gospel?
It may be possible to argue these points to a certain degree from a non-evangelical perspective, but for the vast majority of YEC's the answers are evidently (and respectively):
A1. Unambiguously yes.
A2. It is critically important - our understanding of the gospel would be seriously undermined without it.
Conclusion: An essentially literal* interpreation of the fall of man (as seen in Genesis 1-3) is critically important to our understanding of the Gospel.
edit: Another issue that occurred to me regarding this: if you do not link (the initiaing of) physical death with the (inital) instance of sin, it significantly undermines what sense came be made of the Atonement in particular** - how can Christ's physical death have any impact on sin and it's consequences if the consequences of sin do not produce physical death (but are just the result of natural processes)?
*by this I mean that the 'essentials' of the story (no death before the fall, death comes as a consequence of the sin of the progenitors of the entire human race) are literal, overly literalistic obsession on minor details is not necessary.
**also (OT) atonement more generally - why should we have "without the shedding of blood there is no remission of sins"?