From microcontrollers to microprocessors [Embedded Devices]
I was told by some developer that embedded controllers are task specific devices that requires low latency which processor based devices cant offer due to OS overhead.


That's one reason to use MCUs (microcontrollers). There are other reasons too, such as lower cost, smaller physical size, lower power draw. Think about all set-top boxes, smartwatches, dishwashers, car dashboards, etc.


  microcontrollers without OS


Note that it's common for microcontrollers to run special "embedded" OSes with only the most basic functionality such as memory management, task scheduling and some light hardware abstraction. These are often geared towards giving guaranteed timing, then calling themselves a real-time operating system (RTOS). These OSes are commonly compiled together with the application into a single binary, thus not really resembling the role of a PC OS where applications are separate.

I would say that only microcontrollers with 32 KB RAM or less are often programmed "bare-metal", without any OS library. Embedded OSes are used up to perhaps a couple of megabytes of RAM.


  Can the basics of functionality such as web surfing device be designed purely using microcontrollers without OS.


In this context, web surfing is far from being basic functionality. It requires TCP/IP communication, processing large images, running Javascript, rending to a high-resolution display, etc. While early attempts at mobile browsing was possible on non-smartphones using WAP, and text-based browsing is possible, I'd say in general this functionality is well into the processor domain, using many megabytes or RAM.

If the device has such powerful hardware, there's just no reason to forego the convenience of an OS. Once you programmed your bare-bone system as a browser, you have recreated the OS functionality anyway. That said, I think the OS could be a small embedded OS and not necessarily a full Android/iOS system.

An important factor is how much engineering you're willing to do to push a lot of functionality into a small piece of hardware. It can be year's worth of work to fully optimize the power of an embedded system by rewriting commonly available software just to make it a little smaller or faster. You could buy a lot of more powerful devices for that money.