Generating AC Power

        Hi HW,
    you are right, the transistors act as switches here. One of them is always opened, the other is closed, so the direction of the current in the 12 V winding alternates - in effect it is A.C. ! You want to it without transistors ? O.K., remove them and the resistors too. Use a S.P.D.T. switch instead: the central (moving) contact to ground, both fixed contact to ends of 12 V windings. Now switch the switch to and fro 40 - 100 times per second. This is what the transistors do.
    Never let the central contact stuck to one side - you would get a D.C. short-circuit of the battery and probably some fireworks on your table !! Never touch any conductive part of the setup when powered. Even the 12 V primaries may develop fairly big voltages due to self-induction. The secondary voltage is lethal !!
    You have a pain in your fingers, don't you? Now you know why the transistors are necessary. Before they were invented, electromagnetic buzzers (vibrators) were used to convert D.C to A.C.

Study some relevant theory before you continue your experiments !
Have you already measured the voltages ? What are the results ?

BR from Ivan OK1SIP