I made some investigation with equal problem using a powerfull adjustable laboratory power supply. the M1001 requires approx 4.25V and quite much current.
Just to avoid confusing people - the M1001 requires
5.0 volts. It will work over a small range of voltages (nominally 5.0 +/- 10%). Your unit happened to start working at 4.25 volts, but that is below the minimum supply voltage for some of the parts and can not be relied upon.
You are right that changing the capacitor in the supply could fix it. However there are other parts in the supply, including other capacitors, which would age over time and which may also need replacing to ensure longer term reliability.
As a point of safety, I would NOT
recommend anyone to open up one of these supplies and start replacing parts unless they are suitably qualified, experienced, and fully understand the risks. On European mains voltages (over 240 volts AC in some parts of the UK) the maximum stored voltage on the main reservoir capacitor would be in the region of 350 to 400 volts DC with enough stored charge to cause serious injury or even to kill.
Oooh - my 1000th post.
Almost 4 years since my first.