Because the viewfinder wouldn't work, then. That's the point of a SLR: You see exactly what the lens sees, exactly when the lens sees it.
My emphasis on the
exactly when. You don't get that with live view, you get slightly old data, particularly in low light.
There are DSLRs that use an electronic shutter at speeds faster than the sync speed of the mechanical shutter.
Also remember that the shutter and mirror protect the sensor to some extent during lens changing. Battery life is lower when live view is used.
Some of these reasons apply more to the mirror than the shutter, in fairness.
One main reason is the way in which a CCD or CMOS is read. If readout occurs in the dark, the entire image can be read at once, so motion artifacts are minimised. If the sensor is read while light is falling on it the timing may result in motion artifacts between photosite rows. Mirrors are not lightproof if they allow some light through for focus and/or exposure, so can't be used to guarantee that the sensor is in the dark during readout (unless the secondary mirror flipped up to close the window in the main mirror - but that would be complex).
Even when used in live view, my Nikon closes the shutter to stop live view, opens the shutter to take the exposure, then closes the shutter to read the data, then re-opens for live view. (It also waves the mirror around for no apparent reason...)
Best,
Helen