There can be no one size fits all recommendation for all vintage computers.
For example some DEC power supplies incorporate an over voltage circuit that short-circuited the output of the power supply. If you attempted to apply power with the power supply disconnected from the computer this circuit would trigger. If you made major changes to the internal power supply load in the full sized boxes by removing all the memory and i/O cards in one change you could trigger this circuit. Taking the cards out in stages and adjusting the power supply voltage avoided this problem. The thinking behind this design was a replacement, modular power supply was only worth a small fraction of the total system cost. An over voltage power supply could instantly do tens of thousands of dollars worth of damage in a minicomputer.
As well as looking for signs of leaking electrolytic capacitors, look for capacitors that have changed shape e.g. tops that have slightly bulge up. For an old system performing a visual inspection, assuming all is good then taking the final step of applying full mains power is asking for a loud bang, possibly with a small puff of smoke, the failure of the system to start, with smell of an electrolytic capacitor that has expired, a distinctive smell.
There is a piece of equipment called a variac which allows you to control the mains voltage applied to a system. I would use a variac to very gradually increase the mains voltage while monitoring capacitor and power supply voltages. Depending on the circuits and physical layout this has the potential to deliver a fatal shock. What you are attempting is to reform the electrolytic capacitors. There is a high probability of very old components changing value. Due to manufacturing tolerances electrolytic capacitors tended to be sold with very wide tolerances compare to other components. Good designers assumed the worst and tended to specify electrolytic capacitors rated with a higher capacitance than absolutely necessary for good circuit behaviour. This means you have a reasonable chance of success. A good precaution is to use a Cathode Ray Oscilloscope (CRO) or modern equivalent to check power supply ripple. A Digital Multimeter (DMM) could also be used. Very old circuits may be specified for analog multimeters with an Ohms per volt (Ω/V) and using the wrong meter could give erroneous readings. The problem with a CRO is you may not know what is acceptable ripple and what is unacceptable.
There are two types of integrated sockets used in vintage equipment and the more expensive machined pin type should be more reliable. The type using spring metal contacts can have reliability issues. Soldered joints should be more reliable. Check for hairline cracks on the Printed Circuit Boards (PCB) copper tracks. Dry or bad solder joints can cause problems and may be found during a close visual inspection. Search the internet for any active user groups and investigate known issues. Visual display units contain even more hazards due to the Extra High Tension (EHT) voltage supply and the dangers associated with a Cathode Ray Tube (CRT).