Not quite -- the SG3525 is a bridge controller chip.
The drive transformer looks plausibly like it's one for a BJT H-bridge, which is typical for older stuff (cheaper). This is similar to a MOSFET gate drive application, but less flux is needed, and some of the load current is fed through the transformer as positive feedback. This accounts for base current, and sharpens turn-off. The driver, in turn, has to be able to push around that load current; normally the transformer is driven with a shorting-mode circuit, so that the transistors can be forced off quickly, rather than being left on until they later turn off*.
*While such a circuit is similar to an SCR (latching) driver, it won't stay latched because the drive transformer saturates after a maximum on-time. When this happens, the transformer becomes a low impedance and near-zero voltage drop; this shorts out the B-E junction, turning off the one transistor quickly. Meanwhile, stored energy dumps out of the transformer as voltage reversal, triggering the opposite side switch. Without a driver controlling the transformer, this circuit naturally self-oscillates, as in the above link.
The circuit is probably being controlled for average DC output, but not fast enough to control output ripple, which of course is impossible given the lack of primary side energy storage. Probably they did this, because the light output doesn't need anything -- it can handle the ripple just fine, and filtering and regulating it would be overkill. Rather, energy storage, and therefore cost, can be saved by putting the reservoir only on the circuit DC output side.
Bonus: the primary side ripple gives a high power factor, probably around 0.9 or better (versus a typical 0.5-0.6 for a cap-input rectifier). Maybe not high enough to meet modern harmonics regulations (which need PF > 0.99).
Tim