Can an operating system without api still work?
No, the other programs sit on top of the operating system(OS) and the OS runs on top of the hardware. The OS provides these programs with basic resources the programs will need to run like CPU, memory and input/output from for e.g. mouse and keyboard. The API is the only way they can get these resources, so normal programs will not be able to work on an OS without API. The new browser analogy is wrong since using IE is just a way to download the new browser files(can do it from command line, without IE). New browser still needs the OS to run.