Longer answer: what you are interested in is called the “transmission mechanism”, and the reduced workforce (and also investment by businesses) is only a part of it. The main part of the transmission mechanism is consumption: when interest rates go up, saving becomes more attractive and people shop less, preferring to spend later. They also face lower asset prices (did you notice how all bonds lost value over the past year? that's what it is) and so choose to spend less.
The inflation-unemployment trade-off that you mentioned is also a part of it. All economists know that there are trade-offs between policy goals, and you have to make judgement calls about what the society's priorities are. If only the voters could vote on “2% lower inflation compared to now” vs “1 percent lower unemployment compared to now”. For better or for worse, they don't do that.