Hey fine me me. I don't consider a chip that was designed to do specifically what you asked a bad recommendation or overkill.
Certainly you can use some transistor switching logic... the average cheapo PIC programmer had to solve this very problem as well... just look up some old PIC Programmer designs.
Personally, I'd prefer to use a chip I know will supply the flash with the correct voltage/current with tolerances regardless of Vin (being dependent on 5V Vcc).
Actually Maxim killed off that chip... replaced by MAX1822 If it were me... and I spend all that time cobbling up the 595's... I'm fer sher wanting to pay the $5 for the MAX1822 and keep it simpler
I was going to mention that these parts usually want 5-12 not 0-12 transitions on PGM pin, as mike says below.