Uh bruh. I took this class when I was 22 at NYU. Quadratic variation, brownian motion, and of course black-scholes etc. A lot of the work is based on a Japanese guy named Ito, who pioneered Ito integrals. And yes you need to know basic measure theory or probability as a prerequisite (take Math Analysis at least)
The closest I ever got to being a quant is doing an internship at a hedge fund called Concordia. They were just using Excel and VBA for credit default swaps back in the day. I then ended up at Bloomberg building their front end in C++ which st that time was a huge compiled binary.
I quickly exited that world and realized I enjoy building web applications. Had been doing that ever since. Guess turning $220 billion into $223 billion wasnt my idea of fun.
What you need as the key is Python, ML, SciKit, etc.
Adding to this, stochastic calculus matters more for modeling volatility/interest rates/derivatives. As you mention, Python/ML are more than suitable for many other areas within quant finance like optimization, algo development, signal research, etc.
It depends on where you are - many large banks have their derivatives pricing libraries written in c++ or c#.