Aurora lets you grow your db storage up to 128 TB, but things become difficult and slow long before this point. This is true operationally (e.g. schema changes, restoring from a backup, etc) as well as potentially for application / query performance issues.
That said, sharding can be very painful too in other ways. But at least it permits infinite horizontal scaling, and it makes many operational tasks easier since they can be performed more granularly on each smaller shard.
Besides, once you get anywhere near Aurora's storage limit, you'll need to shard anyway. So it really just buys you some time before the inevitable.
I haven't used Aurora - what's painful about this approach (apart from potentially being locked-in to AWS / Aurora)?