This CO2 DIY method is very well proven, by very many people. The basic idea is not flawed. What does give lots of people problems is tiny CO2 leaks where the tubing has to be sealed - the bottle lids, the check valves, the reactor connection, etc. It can be hard to find those leaks, but you have to find and correct them. Start by using a liquid dish detergent/water mix sprayed or dripped on every single connection. Wait several minutes and look for tiny clusters of very small bubbles. Those are all excessive leaks.
I can't tell how big the bottles are, but anything smaller than one liter bottles is unlikely to give you CO2 for long enough to be useful. Also, the temperature of the bottles of yeast/sugar/water need to be at least at warm room temperature or the production of CO2 will be very slow.