Most tissues in the body can get their energy from burning fat. Others can't burn fat but can burn ketones (a by-product of fat-burning). A small number of cells, mostly in the brain, can only get their energy from glucose. I don't know where white blood cells fit into this picture - perhaps another PH will know.
The liver will release glucose into the blood stream as and when it's needed. Unless you have problems with insulin sensitivity (or production), this is a well-controlled process in which the liver automatically maintains blood sugar levels within their correct range.
The liver can normally, if and when necessary, derive the necessary glucose from the breakdown of proteins and fats. So it's not strictly speaking essential to eat any carbohydrates in order to get the correct amount of glucose into the tissues.
Some people (like me) however don't like having to rely on breaking down fat and protein to get their glucose. In my case, it leads to excessive cortisol production. Instead, I aim to eat just enough starch (which is broken down by digestion into glucose) and/or glucose to meet the physiological requirement for glucose and maintain a bit of glycogen in reserve in the liver.