You want to go take a proof-based college math class. Or sit in on one. Or find videos of one. It is hard to guarantee that a course will be like that, unless its name is either "Analysis" or "Abstract Algebra". Don't sell Indian universities short - it's so hard to get a good job in mathematics that you have to go very far down the hierarchy to find a place with bad mathematicians.
And you do need to memorize techniques - the only issue is that the techniques you need to memorize are at a much higher level than you are currently being taught. Techniques include "induction", "contradiction", "diagonalization", "construction", etc. Mathematics is a toolset, and the techniques and facts you learn are the tools required. They are beautiful in themselves, and can bootstrap to better tools, but if you aren't a mathematician then you should be memorizing solution techniques, just at a higher level than you currently are.
And you do need to memorize techniques - the only issue is that the techniques you need to memorize are at a much higher level than you are currently being taught. Techniques include "induction", "contradiction", "diagonalization", "construction", etc. Mathematics is a toolset, and the techniques and facts you learn are the tools required. They are beautiful in themselves, and can bootstrap to better tools, but if you aren't a mathematician then you should be memorizing solution techniques, just at a higher level than you currently are.