If I'm not mistaken, you can make a hybrid renderer which lets a rasterizer solve the visibility problem, then shade all of the projected shapes with a raytracing algorithm. It seems that this isn't commonly done however, what's the reason for that? Oversight?
Disclosure: I work for IMG/PowerVR
Indeed there are rendering systems that can do this: Imagination demonstrated hybrid rendering on their "Wizard" Ray Tracing/Rasterisation system. Scroll down to "Making ray tracing happen" and there is a screen grab of a hybrid demo.
As others have mentioned, there are pros and cons: Some advantages include that rasteriser can be extremely efficient at handling the first set of intersections and, for shadows, you don't spend effort pre-rendering portions of shadow maps that may never be visible in the final render. On the downsides, you are typically constrained to the standard pinhole camera model.
Rasterization is sometimes used for primary rays. However, it limits greatly what you can do - depth of field, motion blur, participating media, refraction, only basic camera models, no adaptive or custom sampling, etc.
There are renderes that do this. Most notably Pixars Renderman prefers to work this way. Hybrid renderers can work in other ways too...
It used to be that hybrid renderers like this were quite common, they still are somewhat in software only implementations. They are slowly being replaced though by a new breed of trace only renderes though. Mostly the reason is that by foregoing the other you get a simpler piece of code.
Because when you test the visibility of a fragment during rasterization you already know (or can easily calculate) all the details that raytracing will give you.
So doing the ray-triangle collision calculation is redundant.
But it is the additional lighting beyond the first hit where raytracing is helpful. However each ray is only a small part of what contributes to the lighting in a scene, so you need a lot of them per fragment. And each ray has relatively large cost to compute.
Those two properties make raytracing not very atractive for real-time rendering without some dedicated acceleration.