3s power is indeed averaged over 3 seconds, every 3 seconds. Instantaneous power (usually every 0.1 second or less) is mainly relevant to racers when training or racing events where peak power is a measure of training results and will help set training goals for the future. A good example would be a sprinter who wants to know their "jump power", or the first 1/4 stroke to start the sprint. It may only be able to be held for a fraction of a second, but it can win races, and you can train for it, mainly with weights. 3s power is important for sustained efforts measured in seconds or minutes. Beyond that, the gold standard for rider power is FTP, or Functional Threshold Power. The time metric for FTP is 60 minutes.
The three main metrics for training on the bike are power, heart rate, and RPE (Relative Perceived Exertion). RPE is, "how do I feel right now while I'm expending this effort"? Heart rate is a lagging indicator, by at least 10 seconds or more, and RPE can be as well. Power is instantaneous, and averaged over 3 seconds can be pretty accurate.
The average power metric depends on what's measuring it, but it usually is measured in minutes. On my Kiox, it seems like 3 minutes or so. I use that along with RPE and HR to gauge fitness after a ride.
I hope this helps.