Yes weight matters. It matter more on hills and stops and go. On flat once you get up to speed it only has a small effect in rolling resistance. Spinning weight - weight in the wheels matter more - like a 2x.
But it is relative. Take the bike weight plus rider.
From 100 kg to 90 kg is 10%.
From 50 kg to 40 kg is 20%.
For sure going to 7-8 kg will have huge difference in cost. Even on a small frame you are going to need to go carbon pretty all the way (from handle bars to wheels). Even consider a carbon fork over a shock.
I have full carbon frame and carbon fork single speed medium and it weighs 18-19 lbs. Even with carbon wheels I am not sure it would make 16 lbs.