Why couldn't IIM A, B, and C compete with NITIE in operations?

The question you have asked can also be interpreted as to why Microsoft is not currently competing with HAL in producing similar products since they both come with their own personal expertise. 

Either of the institutes does not need to compete against one another because they already come with their own prospects, image, and information. 

IIM Ahmedabad does not actually emphasize students to opt for a management function for pursuing a major or specialization. The course of general management is 2-years long with the liberty to choose electives as per one's choice and on the basis of academic profile in the second year. Students, before placements, are asked to rank and rate companies that they want to call on the campus during future offers. 

In fact, IIM - A, B or C is not particularly competing with NITIE, Mumbai in Operations because they both have maintained their own personal brand values which are gigantically different. 

They have, at present, established a whole perspective of General Management B-Schools. Apart from that, IIMs do not apparently require to participate in a competition against NITIE as any management course since students over and across India generally only prefer the former anyway.