About me: I'm a member of the hiring public. I've worked 35 years in a multination company with a significant presence in India, and a heavy dependence on technology.
What is Value? If "value" is measured in terms of what doors it opens, then you may open more doors worldwide with a US degree, provided it is from a more recognized university. This is merely a matter of people taking mental shortcuts, to rely on more known commodities (MIT vs. unfamiliar school X), and has nothing to do with actual education quality. However, within India, the value of a degree from a well-reputed university or college may well be of comparable value, simply because that recognition bias is less of a factor.
Worth to Employer. In my company, we have many outstanding native Indian engineers working in our Indian arm, most of whom I believe earned degrees within India. They are bright, driven, professional, and can talk tech with the best of us. Often, they deliver some of the best technical insights and process developments in our company. This trend became particularly noticeable somewhere around 2000, give or take a few years, and has been building since.
Pride. If your essential question is about self-respect, then hold your head high, and focus on your work. Once in a career, the quality of your work and shared insights are all anyone every cares about. My degree was from a state university with only modest name recognition, yet I have thrived and my technical contributions have earned wide respect, simply because I asked good questions, worked hard, collaborated well with others, and made the business goal my personal goal. No one cares who issued my degree.
Foot in the Door. An important caveat here is the word now: "No one cares who issued my degree, now." Before I was hired, they needed to filter applicants somehow, and at that stage, yes, for lack of any better sorting information, they did consider my college. But they looked about equally at other characteristics that they felt made an important fit to the job and to their corporate culture (things like work experience while in school, as a measure of how self-driven one is), and of course how well you interact in an interview. For every entry-level job with every company, there is a corporate culture that impacts their hiring preferences. These are impossible to predict as an applicant. So, just present yourself honestly (& positively), and realize that if they pass on you, you may well have passed on them too, if you fully knew them.
Paying Your Dues. It seems common today for new adults to expect to rise meteorically to the top, as if the world were a table set for them alone to come dine, and everyone else is just supporting cast. At least in the US, and perhaps the West in general. Such unrealistic expectations put undue pressure on ourselves. One beauty in accepting "paying our dues," is that it sharply reduces the relevance of one's school. When my company hires a mid-career engineer or scientist, we look mostly at their work experience, and at most only in passing at their school.