Work smarter not harder!
'Shotgunning' all the caps is rarely necessary, takes too long and isn't particularly cost-effective.
Its really quite simple if you have an in-circuit ESR tester. First go around the board making sure all electrolytic caps are discharged (if you want to spend your time repairing the board not the ESR tester!), use a suitable resistor for large caps in the PSU, and simply short small caps. Then test them in-circuit and if in doubt compare with the ESR of a suitable replacement. If any give ambiguous results, desolder one leg, prod it gently to check it isn't resting against the pad or the side of a plated through hole, and check again. (The ESR tester can also be used to check the leg isn't contacting the pad.) Remember the ESR tester cant distinguish between multiple caps in parallel, so again desolder a leg. Its pretty quick to desolder one leg and resolder it afterwards - certainly quicker than completely desoldering the old cap finding the right value, and replacing it properly.
Small caps are the most likely to have gone bad - up to about 1/2" diameter, and also larger caps that are stressed by high ripple currents e.g. smoothing caps immediately after a rectifier on the secondary side of a SMPSU, or are stressed by heat from adjacent power resistors or heatsinks. If any small caps are removed, replace with new, don't put the old one back as you have just stressed the leads and bung seal removing it so the odds are the old cap will go bad fairly soon even if it tested good. If you get a bad one and you cant explain it as excessively stressed, replace *ALL* identical ones.
SMD caps *should* always be replaced if disturbed so if you feel the need to lift a leg, you'd better have a replacement handy.
If you don't have the correct spares in stock, and end up refitting caps or using salvaged ones, for %DEITY%'s sake mark the top of the can *and* the solder joints of the ones you put back with a coloured marker pen, and order the correct replacements. You can soak-test the device while you wait for the parts, then go through it again making sure that everything you marked gets replaced.
For non-electrolytic caps you need to know their failure modes: e.g. the infamous RIFA class X caps that crack, let in a little humidity then belch smoke and flames!, when found across mains inputs and in other high energy circuits should *ALWAYS* be replaced on sight. If any cap shows signs of cracking or surface crazing or discolouration, don't trust it.