The water will indeed cool down when you put the pasta in. Using a lid can help with this though and bring it back to boiling quicker. I usually don't bother with this. And the pasta will still absorb water if it's close to boiling instead of boiling. And since I check visually and by tasting whether it is done it's not that big of a deal. Stirring is part of that. I do that in any case to make sure things don't stick to the bottom.
Anyway, that's just my process. There is no wrong or right here. And you brought up a fair point of the pasta cooling down the water. My mode with this is that a lot of Italian nonnas wouldn't have a lot of fancy kitchen equipment anyway. Like fancy timers or even a clock. A lot of Italian recipes is primarily about good ingredients and celebrating those. Not about tools, techniques, or Michelin star nonsense. So, I try not to overthink it. If it tastes good, I'm happy.
Anyway, that's just my process. There is no wrong or right here. And you brought up a fair point of the pasta cooling down the water. My mode with this is that a lot of Italian nonnas wouldn't have a lot of fancy kitchen equipment anyway. Like fancy timers or even a clock. A lot of Italian recipes is primarily about good ingredients and celebrating those. Not about tools, techniques, or Michelin star nonsense. So, I try not to overthink it. If it tastes good, I'm happy.