Cooled lava looks black, but why the whole volcano, even near crater, doesn't always appear black like cooled lava?
The cooled lava might be covered by ashes. So depending of the amount of ashes and the wind you might have a black volcano or a gray volcano. Many volcanoes are formed by layers of lava and ash.
An active volcanic crater is generally a highly toxic environment full of highly-acidic, sulphur, chloride and fluoride-rich gasses emanating from, and reacting with hot rock. Weave into the mix sudden exposure to oxygen and meteoric water (which may become reactive steam) within the rock, and you have a cocktail of extreme weathering, geochemical reaction and rock corrosion. So the original black rock doesn't stay black for long.
One bizarre and very rare exception is carbonatite volcanoes, such as occur in remote parts of the East African Rift Valley. These 'lavas' start off as white magma, and almost immediately turn black upon exposure to air.
Not all lava is black, the mineral content and cooling speed of the lava has an effect. Cooled lava can have a variety of colors including stark white pumice. https://hvo.wr.usgs.gov/volcanowatch/archive/2000/00_10_19.html