There's a decent amount of content on trying to get a job with system design, so that's a good start. I know I impressed one company because I dropped a sequence diagram on them, there's a few really good UML type of approaches that is worth at least knowing about. Being able to draw a state machine is critical in my opinion (again UML or some format, but it should be clear and understandable. Detailing the states, and decisions as well as transitions)

You probably "Do" system design if you're a senior, it's just you don't write it or review it. If you ever go for an interview and they go "design a X" try to do it with out writing code (at first), and that'll help you.

If you want examples.

Design a car parking lot, that has small, medium and large spots. Cars might be any of the three sizes and are assigned spots as they come in.

Design an elevator where you're trying to maximize the throughput. How about if there's multiple elevators.

Design a stop light. (What about left turn lanes, right turn lanes, what if no one comes in a direction.. what if the sensors broken)... and so on.

Design a rail road crossing, trains come infrequently, but they MUST not be required to stop except in an extreme emergency. In an extreme emergency how would you signal to the train that there's a car on the track.

Here's the thing though, none of these have a "Correct answer" no matter what anyone tells you. System design is about identify and coming up with trade offs. Do you sacrifice speed for safety? Are you maximizing utility? Can you discuss these in terms that make it easy for the other person to understand.

When you are writing official documents it will be more complicated but it's also something you'll have to learn as you go because each company's document will be different, if you at least can design the system before you write code/pseudo code in the meeting, you'll do well.

The bigger problem is you haven't actually done that too much at work. Just to go further, I tried to join a specific FAANG company 3 or four times as a Game dev, Always failed, but obvious I went back so I didn't do THAT bad.

Did 4 years at another company, got in first attempt after that. What changed? Well I could actually talk about my experiences AS a senior. Projects I actually lead, not by just by telling others what to do, but by getting buy in, writing the document, getting feedback, and evaluating the results.

Basically "not coding"... and that's the thing. as a senior, you code less, intentionally. You're designing, architecting, building.

Though I will say the smaller company still hired me and I passed their system design with flying colors, so ... it's not that hard, it's probably more that you're lacking the actual experience, than the actual skill to do it.