I wouldn't worry about the lack of current through the relay contacts. Gold doesn't oxidise, so there's no layer to breakdown, so the contacts will make a good electrical connection, even with the femtoamp current leaking through the capacitor.
It is not the capacitor leakage that provides the contact cleaning current but the discharging and charging currents of the capacitor when it swings from the west side to the east side. The longer the capacitor is held at the east side the more it discharges into the input resistance of the MCU and consequently the more it charges when it swings back to the west side. So there may be an advantage in holding the capacitor on the east side longer than is absolutely necessary for the MCU to read the capacitor voltage.
Hmm, that is interesting about the characteristics of gold clad contacts and would open up a whole new avenue of applications, but are you sure it is true? Do you have a reference for this?
In general, relay (and switch) contacts, are full of gotchas, as I have discovered over the years.
Just in case, there are a wide range of contact materials, ranging from tungsten through silver to gold. But the contacts are never solid gold, they are either gold flash or gold clad (plated).
The gold flash is very thin and not intended to be part of the switching function. Instead it is to protect the contact from oxidization/contamination during storage and before first use. This is especially important for silver contacts. So with gold flash contacts, the gold flash vaporizes on first use and you are left with the contact base material, normally silver.
On the other hand, with gold clad contacts, the gold is used as the contact material for low level signals, but if you use a gold clad contact for high currents/voltages the gold cladding vaporizes and you are left with a contact of the underlying material.
I am not sure that the 10mV/10uA figures on the data sheet for gold clad contact relays is not an implied minimum to keep the contacts clean. In fact, the datasheet for the Panasonic DS relay, linked below implies as much.
The other gotcha with relays is the galvanic voltage, which is variously listed as 10mV, which is quite considerable if you are switching low level signals (the 0 to 150V on the west side could be at a very low voltage). That, burden, and contact cleaning, are the reasons why I originally chose to switch at the full 0V to 150V rather than 0V to 5V.
Relay manufacturers are a bit reticent about the details and performance of their contacts, and I never did get around to establishing the actual position with 'gold' clad contacts.
https://www.mouser.co.uk/datasheet/2/315/mech_eng_ds-1299285.pdf