Do you have to do Nintendo poweroff every time?
If by "poweroff" you mean turning off the console completely, rather than simply putting it to sleep, probably not.
I couldn't find any explicit information on what exactly affects the time limit, but from the little I could find, it seems that the time limit affects the entire console for as long as it's not suspended.
According to Nintendo:
You can set limits on how long or how late the Nintendo Switch system can be used each day. When the time limit has been reached, an alarm notification will pop up on the Nintendo Switch screen.
The app will also tell you if the Nintendo Switch console stays on past this time limit. If needed, you can always set the system to automatically interrupt when the time’s up.
Parents can disable play time alarms and game suspension features temporarily or for a full day using their PIN. For more details about how to use Nintendo Switch Parental Controls, visit support.
The wording appears to imply that the time limit affects the console, rather than just games.
This is further supported by the following Nintendo Support article:
What happens when the daily time limit is reached?
When the daily time limit is reached, you can choose to either have the system suspend the game or simply have a notification appear on the screen.
If the game is suspended, the system will go into sleep mode immediately, and the player will not be able to continue their game. The system cannot be played again until the restriction is lifted or the daily time period has passed. Players can usually resume from the same spot once the restriction has been lifted.
"If the game is suspended" is the most explicit piece of information I could find. If the system is put to sleep while the game is suspended, then that would mean the time limit can be reached even if the game is not running.
With that said, I think this is a good opportunity to teach your child some good habits, such as turning off devices they don't use (or at least putting them to sleep instead). This way the battery won't drain as fast, won't need charging as often, and won't deteriorate as fast.
Note that putting the console to sleep is a very different process from turning it off. Turning the console off would require pressing the POWER button for at least 3 seconds, then selecting "Power Options", then "Turn Off". Doing so will also close the game (if one is running) and any unsaved progress will be lost. There are very few situations where you might want to turn the console off completely:
Saving as much power as possible. The Switch consumes almost no power when turned off, as opposed to consuming very little power when in sleep mode.
The console isn't behaving properly. Turning the console off allows it to reset its state. This can help when the console is in a weird or corrupted state.
The console installed updates and needs to restart. When this happens, the console offers to restart for you, so you don't need to manually turn it off and back on.
The Nintendo Switch, being a portable console, is designed like a smartphone: instead of turning it off, you're supposed to put it to sleep. Putting the console to sleep and waking it back up (and resuming the game afterwards) takes very little time.
Putting the console to sleep is as quick and easy as pressing the POWER button once. To wake it up, simply press either the POWER or Home button, then press the A button to resume the game where you left off.