would a knackered PSU still put power across the system?
Yes. Just not enough. The high pitched noise is probsbly not a good thing anyway. If it where me, I'd screw the warranty and get a new high quality PSU. Like one from AC Ryan, Thermaltake or Corsair - all three companies offer high output modular power supplies that help reduce cable clutter in your case. PSU's that come with cases don't tend to be much cop, in my experience.
Much like stock coolers, you usually just get the cheapest solution that could possibly work.