Why do we not try to describe $4$d quantum gravity with a $3$d CFT?

It is crucial to understand that in the $AdS_{5} times S^{5}$ realization of holography spacetime is percieved as ten dimensional by any bulk observer because the radius of $S^{5}$ is macroscopic (much larger that the string lenght). Then $AdS_{5} times S^{5}$ is not defining 5-dimensional quantum gravity by means of a SCFT.

All the known consistent examples of holography are constructed as the near horizon geometry of some piles of branes in string theory. The resulting bulk geometries are always of the form $AdS_{d} times (something)$, always with ten or eleven macroscopic dimensions and never of the form $AdS_{d}$ (such as the case of pure gravity in 3d) or $AdS_{d} times K$ where $K$ is some compact geometry as needed to describe 4d quantum gravity.

In that sense, $AdS/CFT$ is not a useful phenomenological scenario as the usual string theory compactifications are. A parametric separation between the KK-scale and the Hubble scale is the most basic requirement for phenomenology that is abscent in holographic setups.

Update: I recently learned that the path integral of three dimensional Chern-Simons theory can be viewed (under some hypothesis) as equivalent to $mathcal{N}$=4 SYM in four dimensions. Reference: A new look at the path integral in quantum mechanics.