The new method of fishless cycling is actually faster than the add-ammonia-every-time-it-drops-below-1 ppm method. That's because with the new method nitrite can never go above 15 ppm, whilst levels way in excess of 15 ppm are common with the older method and that slows the cycle right down. Nitrite above 15 ppm inhibits the growth of the nitrite eaters.
100% new water won't affect the cycle as the bacteria you want to grow live attached to surfaces. There are virtually none floating in the water.
The nitrogen cycle is this.
Fish make ammonia, it is their version of urine. In a cycled tank there is a colony of bacteria that live on the surfaces in the tank, but lots of them live in the filter because the media is designed to have a huge surface area. These bacteria use ammonia as food, and their waste is nitrite. There is a second colony of bacteria living on all the surfaces in the tank which eats nitrite and their waste is nitrate. In a tank, there are no bacteria that eat nitrate so we do water changes to remove it.
The nitrogen cycle is fish -> ammonia -> nitrite -> nitrate.
A new tank has none of these bacteria so we need to grow 2 sets of bacteria, one to eat the ammonia made by the fish and the second to eat the nitrite made by the first bacteria. This process of growing bacteria is called cycling.
When we add ammonia, the few ammonia eating bacteria in tap water can get started multiplying right away. They multiply by splitting in two every 12 to 24 hours. Because there are so few it takes a long time to make any impact on the ammonia. The splits make more bacteria like this - 1 to 2, then to 4, then to 8, then to 16, 32, 64, 128, 256 etc. As you can see, once they start multiplying the numbers build up quickly but it takes a few weeks to make billions of them.
The nitrite eaters can only get started once the ammonia eaters have made some nitrite, and they multiply slower than the ammonia eaters. So ammonia starts to drop after a few weeks but nitrite takes longer to drop.
I fishlessly cycled a sponge filter this spring using the new method. The ammonia reading did not change until the 28th day after adding the first dose, and the cycle finished on day 48.
If fish keeping teaches us one thing, it's patience