The vast majority of ancient statuary looks incomplete because it is. The missing details used to be painted on. Given a millennia or two and the paints have largely been stripped away, sometimes intentionally by well meaning curators and the like who thought the remaining patina was simply weathering.

We tend to think of Greek and Roman statuary as being blank white marble like this bust of Caligula instead when it was first completed it would have looked something more like this reconstruction.

We know that the ancients painted their statues because they talked about it such as this line from Euripides, Helen of Troy, "My life and fortunes are a monstrosity, partly because of Hera, partly because of my beauty. If only I could shed my beauty and assume an uglier aspect the way you would wipe color off a statue." Or this comment from Pliny , “When asked which of his works in marble he liked the most, Praxiteles used to say: ‘Those to which Nikias has set his hand’—so highly did he esteem his coloring of the surface.”

We can also make some guesses at what the paint might have looked like due to UV, IR, and x-Ray spectrography. This UV photo of a helmeted statue shows what these statues look like in alternative wavelengths. A lot of the paints left traces that aren't visible to our eyes, suggesting what the painted pieces might have looked like, and mostly they were apparently very garish to our modern sensibilities. Like this greek statue of a Persian archer.