The issues when measuring in-circuit are:
If the circuit is powered or has capacitors that remain charged and this exceeds the input voltage threshold then you may damage your LCR/ESR meter. Only test when the circuit is fully isolated from the power source and you have confirmed your capacitors are discharged.
Parallel capacitance and resistances will affect the measurement, so it's not always possible to measure in-circuit.
The LCR/ESR meter may turn on semi conductors depending on the voltage it applies to the device under test, this can additionally throw out measurements.
When checking the ESR health of a capacitor Kelvin measurements aren't so useful. Testing ESR is more a ballpark test. It is dependent on the test frequency and should be ideally compared to the datasheet or a known good capacitor measurement, same series and value etc.
For accurate milliohms or microohms Kelvin measurements are more useful. You should verify by testing out of circuit anyway if you find suspicious components.
Some other tips:
For something like a 30 year old piece of test equipment, if you found some marginal components and you were restoring it for your own use then you would be wise to go through and replace at a minimum all the same values and series that are reused throughout the device.
If the capacitors show signs of electrolyte leaking then you risk damaging the PCB leaving them or the residue for a prolonged period on the board. Capacitors can fail open or short or anywhere in-between and can cause damage to other components. They also can become reduced in capacitance or leak capacitance, so keep in mind it's more than just ESR.
If you are working on a circuit where you can determine the failures are design, heat or stress related then you can opt for just replacing the affected capacitors rather than go for the full restoration approach.