Hydraulics and electric system are independent in commercial aircraft – hydraulic pumps are directly driven from the engines, as are electrical generators. Redundancy is provided via independent loops/buses from each engine. A bird strike on its own is unlikely to be energetic enough to sever one of those independent systems, let alone all four. Losing both engines could do it, – but again, they had enough thrust to attempt a go-around, so they weren’t a glider immediately after the bird strike. The 737 is an old-school design, too, so most critical components have full manual reversion – as long as you have airspeed and altitude enough to get to the runway, you can fly and land the plane just with cable controls and manual releases in the event of total electric and hydraulic failure.
I did a bit of reading from other sources and this particular aircraft predates the requirement for battery backup of the FDR and CVR, and the APU does not start up automatically on a power failure, so the failure chain for that part of the incident isn’t as long as I initially thought. Still, lots of questions, and I think the simplest explanation so far is the aircrew panicking and making a survivable situation into a bloodbath.
737s don’t have RATs. According to some 737 pilots I’ve seen commenting, the APU is operable in flight, but doesn’t kick in automatically and would have required ~60 seconds to start. The main electrical generators don’t automatically restart after tripping, either, so a scenario where electric power is hypothetically available, but a panicked or overloaded flight crew don’t take the steps to bring it online, is plausible.