Will the PSU still power on without a fan directly attached to it
probably not, just like the cpu fan header on the motherboard if there is no current passing thru, the detector will most likely auto switch off
i have no idea how loud your psu fan is but with out a measurement i can't tell you if there are better fans. alot of it is about air flow in cubic feet per minute, if you don't pass enough air thru it, it doesn't matter how loud or quiet your fan mod will be, you'll fry your system at worst or it will keep auto switching off when overheat is detected. assuming you have a good psu and not a cheap one.
lack of specifications holds up any decision making