I have a question about how the circuit of TV display works. Does current flow through the gate lines (horizontal lines)?
The FET gates loading the horizontal select lines effectively detect the presence of a voltage, drawing only a leakage current. So the lines do pass current but not a signalling current. With many thousands of FETs on the line, this will add up.
Each gate also presents a capacitance to its horizontal select line and this needs to be charged up or discharged when the line transitions low-high or high-low. So a higher current will be drawn/sunk until this capacitance is charged/discharged on a transition.