@Dhruv97
Congratulations on your admission, sorry you were not accepted into CSE.
We get asked this question a lot - I mean a WHOLE LOT.
And the answer is this - Math-CS is not a CSE degree, it is not a hardware degree, it is not a software degree, it is not a programming degree. It is a MATH degree. It focuses on the foundational mathematics that underlies the historical development of Computer Science. It is a degree in logic, algebra, combinatorics, probability theory, discrete math and algorithm design.
If you are interested in Math and how it relates to CS, then it’s a great major.
But if, like you write, you are ultimately interested in becoming a programmer - it’s probably not a great major. There will be a lot of courses you’ll not be invested in.
And that’s the main thing - if you land in Math-CS because it’s the ‘next best’ major, and you don’t really dig the math. Then your looking at 4 years of struggle and resentment. That’s not what you want to look forward to going into college.
But, could you be a Math-CS major and achieve the goals you state - most certainly. But it will not be because of the major - it will be because you take the time to go beyond the requirements of the program. You’ll need to develop programming language skills on your own (either additional CSE courses, or self-study, or online, or UCSD extension), join engineering professional societies, seek out internships and round out your CS skill-set. Your Math-CS major will not keep you back, per se, when applying for jobs - any degree in mathematics will show employers that you have developed any number of critical thinking and problem solving skills - but it alone will not provide you with the specific programming skills and abilities you’ll need in Silicon Valley as a developer.
Now - as to why you might consider a degree in Math-CS when what you really want is CSE…
How about this - by the time you graduate, the programming languages you learn in first couple of years at college will probably be considered old-fashioned…the hardware you work on will be considered last-generation…but the mathematics that form the basis of the field will remain unchanged. You will develop a deeper understanding of why CS works - and you can always just add the skills on top of that.
Hopefully that helps your decision - if you have any further questions, just ask…