Your question seems to have two parts. The current is a concern because electrons move slowly and the current is large. It may also be a concern that the light travels so fast, whereas it takes electrons hours to travel from the switch into the light.
The current is the first concern. It depends on the speed of electrons, but also how many electrons are moving at once. The electrons in a wire move only a few millimetres per second for a given time. However, those few millimetres are filled with a lot of electrons.
Let’s say that a copper wire has a diameter 2 mm and is round. This would give a cross sectional area of about 2 mm. If electrons move at a speed of 1 mm/s, then there is only one electron per second.
Although it seems very small, each cubic metre copper contains 8.5 times 1028m3 electrons. C/electron is 40 C/s, or 40 Amperes. This is a large current.
You can have huge amounts of charge moving, even though individual electrons aren’t going very far… there are so many electrons.
Why is it so long for the light to turn on when electrons move so slowly? It is because electrons travel slowly from the switch to the light, which makes it seem like they are moving. There are plenty of electrons already in the light… the light must make these electrons move to turn it on.
It is not the electrons moving, but an electric field created within the wire that causes them to move. The electric fields create forces that cause charges to move. Therefore, current begins as fast as the field spreads through a wire (closer to the speed of light). The potential difference between the two charges is greater, so the charges move faster. Ohm’s law states that higher voltages create more current. ).
It is still too slow to spread the field. High speed applications. However, light switches are not as popular!