Wow, you have an old timer from Corsair in use.
😀 I too have Corsair Obsidian case where my Haswell build sits in but mine is 750D Airflow Edition (full-tower ATX).
Since now i have access to the case specs, that last button in row on your front I/O (round one) is the reset button. Also, your case does have hot-swap bay at the top of the case as i suspected. Hot-swap bay is mainly used for external drives to back up data. And we also know now what that magical "black box" is in your front I/O. It's the fan controller unit.
650D fan controller spare part:
http://www.corsair.com/en-us/obsidian-series-650d-fan-controller-kit
When looking your images, i thought that the fan controller switch is actually a clip that holds the foldable front I/O cover in place since my 760T also has two of such clips to hold the front grille in place.
Did read from the net that many people have issues with the 650D built-in fan controller. Since fan controller is connected to the PSU and there's a small PCB inside that "black box", it very well may be the source of your burning smell. Unplugging fan controller from PSU would left it inert but all the case fans that are connected to it won't work either.
As far as new PC case goes, you can either go for the one that also has built-in fan controller (e.g Corsair 760T v2 or 780T) or which doesn't have one while having more-or-less the same looks as your current case (Corsair 450D and 750D). Even when the case does come with built-in fan controller, you don't have to use it and left it unplugged, like i did with my Skylake build's 760T v2.
Since i don't know your taste about PC cases, i can't suggest you one but Corsair and Phanteks PC cases have solid build quality. NZXT cases are also popular nowadays.
Corsair lineup:
http://www.corsair.com/en-eu/pc-computer-cases
Phanteks Enthoo lineup:
http://www.phanteks.com/Enthoo-Series.html
Phanteks Eclipse lineup:
http://www.phanteks.com/Eclipse-Series.html
NZXT lineup:
https://www.nzxt.com/categories/cases
Yes, your amazon link shows different kinds of fan controllers, both internal (e.g Silverstone) and external (e.g NZXT Sentry 3). I prefer using external fan controllers that sit in the 5.25" bay since with them, i have more control over my fans and they also provide some eyecandy. And thanks to this, all my 3 PCs: Skylake, Haswell and AMD have external 5.25" bay fan controllers (full specs with pics in my sig).
Internal fan controllers are handy if your case doesn't have external 5.25" bay or if you don't want to see the fan controller. Though, most internal fan controllers are more like fan splitters than controllers since you can't control individual fan and they all run in sync (e.g Silverstone). Only those internal fan controllers that have dedicated software to control them have control over each fan separately (e.g NZXT GRID+ V3 and Corsair Commander Pro).
While all external fan controllers have dedicated channels for each fan and you can manually control the speed of each fan. Many external fan controllers also come with temperature sensors that you can spread all over your case to see the internal temps of different components. Also, some external fan controllers have LED touchscreen if you like it more than knobs/switches (e.g NZXT Sentry 3 that i have in my Haswell build). And some external fan controllers even have support for LED strips (e.g Thermaltake Commander F6 RGB that i have in my Skylake build). Of course, you need to read the external fan controller specs (and review) before getting one since some may come with different view angle. E.g Aerocool X-Vision that i have in my AMD build is designed for PCs that sit below your eye level, making it easier to read their info from above, rather than straight on.
When i was looking for a fan controller for my Skylake build, i made a topic where i listed many fan controllers with links to the specs. You can use my topic as a source for the specs of different fan controllers. It's better than searching fan controllers from amazon,
😀
link to my topic:
http://www.tomshardware.co.uk/answers/id-3380313/led-fan-controller-external-bay.html
So, you checked out my Skylake build and saw that i have 7 case fans in use (mostly 140mm) and thought that my PC should be noisy due to having so many case fans, right?

Wrong.
The trick is that the more fans you have inside the case, the less each fan has to work to maintain the airflow and the less noise fans produce. Also, 140mm fan movers more air than 120mm fan at the same RPM. And that is also a main reason why i have 7 case fans in my Skylake and Haswell builds. Since i have that many case fans, i can keep all of my case fans spinning between 800 - 1100 RPM and thanks to this, my PCs are very quiet while still having proper airflow inside my full-tower ATX cases.
Of course, i'm using high airflow and high static pressure fans inside my PCs since even when they spin at about 50% of their max speed, they still provide good airflow.
If i sit still and do nothing then i can barely hear my fans working but the sound they produce is low and calming. If i'm working on something or listening some music, i can't hear my case fans at all. Closest sound example to my fans is this river flowing sound but my fans are much quieter than that sound in the video,
youtube: https://www.youtube.com/watch?v=eRFzN_x_pww