Other commenters have pointed out the problems with overloading of connectors and reduced efficiency because of the added resistance but there is another really important reason not to chain power strips: circuit breakers work best against short circuits when the resistance between the breaker and the short is fairly low (for instance less than 0.5Ω) so that the current will quickly go over the rated current of the breaker. If the resistance is a lot higher because you have too many extensions between the breaker and the fault, the time the breaker needs to react will go up. Counterintuitively this usually means more energy will be turned to heat by the fault.

In extreme cases this can mean the difference between a broken power strip that you can just throw out and a burned down house.

skipped reading the silkscreen … it says Q1 on the lowermost package but the other three are diodes. so could also be a forward converter or something similar.

indeed it looks like a Royer oscillator in there (because there are transistors before AND after the transformer).

Acording to the datasheet the TDA7294 uses -V_s and +V_s in the block diagram so i would assume it is intended to be used with DC power. If the module is specced for use with AC as well as DC, then this just means what you already suspected: it has an integrated bridge rectifier and most likely some sort of low pass for the rectified power (read: a bunch of big capacitors).

You could just go with a big transformer core that powers them all at the same time; many commercial amps do that and it works fine in general, provided you have enough margin for power spikes and the modules will not influence each other when connected in parallel to power.

In my opinion using separate transformers would be paranoid but it would work of course.

Edit: dont forget that this thing will produce heat. If you really go with an 800W transformer then you have to be able to cool about 400W in the worst case (going by the data sheet power dissipation of the chip and assumed transformer + rectifier efficiency of 90%).