No, the problem is that according to
this silica aerogel is highly transparent, which means that for most visible wavelengths, most of the light will go right through silica aerogel and irradiate whatever lies behind it, and only a small portion will be absorbed and scattered by the aerogel itself. Some of the light that went through will be reflected back to the aerogel, but it would be reflected if there was no aerogel layer as well.
When you see impressive photos like the one below, you have to remember that in this case most of the heating is done through convection (flame is basically a hot gas), not through radiation, and you can clearly see the flame through the layer aerogel.
In practice however, things are more complex. There are
different variants of silica aerogel. Some are nearly transparent, others are translucent. We are interested in the latter variant since it protects the components behind it better, but still 50% to 60% of the light will get through according to the article above. Another
article lists 88% and 72%. But that's sunlight (that contain many different wavelengths) but what about coherent light? Well, according to what I was able to
find, it would indeed be opaque to ultra-violet lasers, but let through most visible wavelengths.
Then, there is there is one more effect that may be able to increase the effectiveness of silica aerogel to the point of it absorbing all laser light:
total internal reflection. However, there are problems with that too. For it to work, we need that the refractive coefficient of the material behind silica aerogel be lower than aerogel's refractive coefficient, and we know that the latter has a very low refractive coefficient already. It would only work if the light is incoming at a certain angle.
Sorry for wordiness.