Question about how the circuit of TV display works
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.