The power split device isn't an ordinary transmission, it's a set of planetary gears with a fixed gear ratio between three shafts. One goes to the wheels, the other two to the engine and the electric motor respectively. The ratio of the engine speed to the wheel speed is then set by the speed of the electric motor connected to the third shaft, which gives you a CVT with no belts, clutches, torque converters or even synchros.
The transmission is "more complicated" only in the sense that it contains electric motors. In every other respect it's simpler, more efficient and more reliable than an ordinary transmission. Meanwhile those electric motors mean you don't need a starter motor or an alternator because the engine can be started by the electric motor through the transmission and an electric motor is a generator when operated in reverse.
A series hybrid still requires you to have a gas engine with all that entails, but now the gas engine needs its own dedicated electric generator/motor and you can't deliver power from the gas engine directly to the wheels, so the traction motors have to be bigger in order to supply 100% of the torque used in acceleration instead of the gas engine and electric motors both contributing. That makes series hybrids heavier, slower and more expensive, so they're basically useless. Probably the main advantage would be that you could offer the generator as an option on what would otherwise be a full electric vehicle and then only people who need the extra range would pay for it.
In practice they also allow the engine to run at higher speeds under heavy acceleration because the peak efficiency RPM and the peak power RPM are different and the assumption is that if the driver is stomping on the accelerator they want to resolve the power/efficiency trade off in favor of power right now, which is another thing the series hybrids can't do.
And of course, there are plug-in hybrids: https://en.wikipedia.org/wiki/Plug-in_hybrid
It's a lot easier to add enough batteries to match the range of an ICE car. Range anxiety is largely manufactured at this point. The cars know how far they can go and where the chargers are. A gasoline powered generator would be a huge extra cost with no real upside other than averting a non-problem.