Look at your tcas and tell me there's 5 mile separation into JFK or any major American airport.
Depends on what the runway usage for the day is. Landing and departing the same runway is not uncommon even in the US.
If tower are departing on the same runway as arrivals...then 5nm gives them enough of a gap to get a departure away.
The visual helps traffic run smoothly when running parrallel approaches (Which alot of airports in the US do). You can have aircraft side by side on their respective ILS`s without having to worry about separation from the other aircraft...separation being the aircraft seeing the field and their parralel traffic.
In non visual conditions you cannot do this, aircraft have to be established on their respective approaches with the prescribed separation (in this case being vertical 1000ft). This will take extra vectoring to ensure aircraft intercept at the correct glideslope point that will provide separation from the parralel (aircraft at 4000ft will have to be vectored to the 5000ft intercept point to stay clear of the adjacent traffic).
It often means missing a gap on final as you can`t legally get an aircraft established without losing separation.....therefore it can`t move as quickly.
When running single runway ops ie land one depart one...I can`t see how flying a visual would speed things up that much as unless you forgo wake separation it won`t make that much difference...infact most controllers should still exercise positive control on you even on a visual especially if you are running 2.5mile spacing or something close. Nothing worse than an aircraft on the visual putting out the anchor to keep behind the one in front..and then the concertina effect starts behind. Much easier to just be packed in with efficient vectors and speed control and the controller then has an idea of who is doing what.
Next time you do a visual at a US airport check to see if they are running parralel approaches, landing and departing the same runway.
Hopefully that gives some clues as to why you are asked to do what you are.