Not sure how your Lumia 620 does it but no matter how hard I press the shutter my Lumia 520 doesn't take a photo without focusing however even if it did, much of what you've said would still be wrong.
Fixed focus camera sensors have never been known to provide very amazing images, they often lack in detail one way or the other but that's not the only thing. I used to have an HTC Pico which had a fixed focus sensor, now it should've taken photos instantly right? Since it had no focusing to do but the phone still had a considerable shutter delay, similar to what is present on a lot of Windows Phones(but not as severe).
But back to your point, when you force the camera to take a photo faster by pressing the shutter button all the way the phone bumps up ISO and let's in less light to hit the sensor, so this method of yours to take "unfocused shots" would essentially mean taking mediocre shots(compared to your camera's full potential). The longer you give the camera to stabilize and focus the more light gets into the sensor and it has to use less ISO, so you would get more detailed shots, this is especially true for low-light photography.
After HTC Explorer I got Xperia S, which had a stellar camera for it's time and it utilized the single-shot focusing method and when I compared it to my cousin's 4S the 4S would consistently take faster photos. I could deny the fact all that I wanted, I would often justify my camera performance stating that my Xperia S had a 12 megapixel camera and give other reasons as well.
At the moment I believe iPhone is probably the only in the industry utilizing continuous focus however most Android OEMs have reached a point where they have perfected speed in their top devices, using different methods. The Snapdragon 800(and 801) being used in most high end devices today has the ability to take photos upto 21 Megapixels, so what different OEMs do is:
HTC One has 4 Megapixel camera so generally the processor takes very fast photos, because of the low res camera. Samsung Galaxy S5 and LG G3 have dedicated sensors(Isocell and Laser) to ensure faster photos. Nokia is the only one behind, we have phones equipped with very amazing cameras however no dedicated software or hardware to get the speed required to take better photos.
The iPhone 5S does indeed deliver better performance in the camera due to many reasons but mainly because of the lower megapixel count and the continuous focusing.
The more detailed reason for this is that Lumia Pureview cameras come equipped with OIS and like letting in as much light as possible so if you have shaky hands, Lumia is not the place to go. The phone clearly favors a slower shutter instead of a faster one. I've managed to take some pretty stellar photos even with my low-end Lumia 520 but most of them were due to tweaking before taking a photo, I could help you if you want some tips in mobile photography.
Nokia still has a lot of work to do, first of all they need to increase the Nokia Camera's startup time, last I checked Lumia 1520 takes around 1 - 2 seconds to open the camera, and I'm pretty sure it takes a few extra seconds to take the photo. Whereas in iPhone the camera opens in around 0.5 to 1 second and is ready for taking lightning fast photos maybe in about 0.5 seconds more.
However with the Cyan update I would imagine the situation will improve drastically but sadly most of those updates are only coming for the recently released high end Lumia range, but I believe maybe with Cyan Lumia 1520 may be able to outperform the iPhone 5S in photography. In the current state, I don't think so.
I would also like to point out the lowest ISO 1520 can reach is ISO 100 whereas iPhone can reach ISO 32, I hope Cyan adds a lower ISO, I hope.
I've really had fun arguing with you. You've raised some amazing points. I think we should keep in touch on twitter.
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