Sorry to hear about your troubles. When you get the kinks out, they should work fine, but it looks like AS's quality control still has a way to go.
AS was bought by the owner of INAX, which is Toto's big competitor in Japan. INAX had tried to penetrate the US market on its own, and realized what a challenge it would be. So, instead, they bought AS, and should bring Japanese quality and technology to the AS line, while continuing to use its established distribution channels, like HD. So, this AS/INAX version of the Toto Double-Cyclone, with some added bells and whistles (i.e. the unnecessary two flappers), is an example. It's a step up for AS, but getting to the point that quality and reliability match that of Toto (or the AS of decades ago) is going to take some time. Toto has a big manufacturing operation here in the US, and they have been at it for some time now, and just plain have the whole thing down now, from design to manufacturing to quality control to customer service. What they don't have is a significant big box sales channel, although you can order some Toto models from HD for delivery to the store or your home (at completely-bizarre prices: they almost match the big internet retailers on some models, and charge crazy-high prices on others, and don't offer the whole line).
PS Beautiful doggies.
PPS If one flapper sticks open, all that's going to happen is that water is going to run through the bowl and down the drain. Your water bill would eventually take a hit, but unless the bowl was clogged when you flushed, it wouldn't overflow.