A large part of optimization is understanding the hardware architecture in detail and then making your software architecture mirror that hardware architecture as closely as possible. A compiler can't do this for you. Much of "performance engineering" is applying your understanding of how the hardware components work and are connected to the software design.

Most software introduces a large number of unnecessary stalls where one part of the hardware is waiting on or bottlenecked by a different piece of hardware. Optimization is often about removing or minimizing these stalls to the extent possible. But to do this you need to understand why specific code choices cause the hardware to stall in the first place. The most basic form of this is CPU cache locality but there are many levels.

Compilers are great at localized micro-optimizations, except for SIMD. Most idiomatic code can be detected and transformed by the compiler into something that takes advantage of the hardware design. You don't need to do tricky bit-twiddling stuff, the compiler is better at it than you are in most cases (this wasn't always true). I always recommend people interested in optimization experiment running snippets of code through the compiler using godbolt with optimization turned on. You can learn a lot about how the compiler sees and understands your code by studying this. It is also a good way to became familiar with assembly.

Leveraging SIMD and vector ISAs is a mess. Different hardware architectures rely on different idioms and compilers cannot auto-vectorize most code that can be vectorized. You need to learn the idioms of each vector ISA and how to write them using intrinsics. A great way to learn this is reading other peoples' SIMD code. Modern SIMD code is wide in that you need to memorize a lot of things but conceptually pretty shallow. It mostly comes down to learning the idiomatic tricks and gadgets for an ISA -- code is composed from these conceptual primitives.

These are concepts I've come across in books I've read, and your explanation is easy to understand. But I can tell my understanding is lacking in some parts because I haven't actually tried it in practice. As you said, I think using Godbolt to disassemble things myself is the right approach.

Thanks for the advice.