GF
Surely the offer of/acceptance of a visual approach fundamentally means that the pilot controls the separation on the approach thus absolving the controller of the responsibility for separation.
It may be that the pilot generates a greater separation than the controller could theoretically achieve.
Now with TCAS etc, it may well be that the pilot targets a 3.5nm comfort separation rather than a 2.5 or 3nm minimum separation - this would slow down landing rates rather than make them quicker, no?
I can see that with local operators happy to wing it with their colleagues at airports with less heavy traffic/single runway airports that it might increase landing rates and likewise at airports where parallel runways are available, where tighter formations are used.
Overall though, I don't think you can state that landing rates will always increase with visuals.
I am now retired, but have operated into many US airports, including some of the busier ones (SFO, LAX, JFK, ORD, MIA etc) so believe I can comment with some knowledge.
Indeed, I was one of those 'furriners' happy to accept visuals into the above, and have accepted a visual onto SFO 28R in a 747-400 with an instruction to 'not overtake' a 757 onto 28L. With a final approach speed of 150kts vs his ~120kts it was very difficult! We landed at the same time at whatever the runway separation is (not much)!
Made me think twice before accepting the visual approach at SFO again (I did occasionally!)
As I said, I think it is more about the controller passing responsibility to the pilot.
With regards the missed approach, then yes, it was the visual missed approach procedure that applied - usually climb straight ahead on runway heading to 2000ft contacting ATC for instructions ASAP.