Aerogel doesn't have much mechanical strength. Orbital velocity is say 7800 m/s and you have to decelerate over say 300 seconds. Even if it's done carefully there are 2 or 3 gees of peak deceleration.
So imagine your spacecraft sitting on the ground resting on its aerogel heat shield with two or three more spacecraft stacked on top of it. It will likely be instantly crushed into aerodust.
I don't know the details of the highest melting point silica gel possible but silica aerogel is said to melt at 1473 K or about 1100 C and that seems to be in the right ballpark though a bit lower than the temperatures required for some areas of the Space Shuttle for example.
The heat shield also needs to be opaque to near infrared light because the plasma sheath is really hot and radiative heating of the heat shield by the sheath is a big problem. Ablative heat shields generate a gas layer that is opaque to infrared and ceramic tiles are opaque to this as well. If the aerogel was transparent to even the shorter IR wavelengths then its thermal insulating properties wouldn't be so helpful.
As velocity increases, both convective and radiative heating increase. At very high speeds, radiative heating will come to quickly dominate the convective heat fluxes, as convective heating is proportional to the velocity cubed, while radiative heating is proportional to the velocity exponentiated to the eighth power. Radiative heating—which is highly wavelength dependent—thus predominates very early in atmospheric entry while convection predominates in the later phases
See also What aspects of reentry heating 'scale as the 8th power'?