Electronic – Does a capacitor connected by only one lead have any effect on a circuit

The solution is not to use the wall switch to control the fan, but the remote, as the product is designed; and to produce a wall holder for the remote which is placed such that people coming into the room are guided to use the remote rather than the switch, and such that they can invoke the "light on" function of the remote without removing from the holder, so that the remote acts as the de facto wall-mounted light switch for the room.

The issue is clearly that when you walk into the room, the main switch is always in its expected location, which reinforces the behavior of reaching for it, whereas you have to look for the remote. If the remote is in a holder that is more conveniently reachable than the switch and as easy to operate, that problem will solve itself via user behavior modification.

You could also make modifications to the wall switch to make it less convenient to operate. For instance, if it is a large, flat rocker switch, build a cover for it so that the rocker can only be operated through finger-sized holes in the cover, discouraging regular use.

If these Hunter Fan people were more clever, they would sell or include junction box covers that double as remote control holders. That way users could uninstall their light switch (wiring the circuit so that it is permanently on) and replace the switch panel with the remote control holder panel, so that the remote control now appears exactly where the switch used to be.

For most practical purposes electrolytic capacitors can be considered to NOT have a minimum voltage rating.

The "problem" in your circuit that causes it to not work at less than 7V is that you are using a 12V relay as seen here. This is designed to operate to specification at 12V applied to coil but will pull in at somewhat less than 12V. How much less depends on various factors including the rate of rise of the applied voltage. Without knowing what is "out there" on your +12VDC supply it's not possible to give a full picture (and even then one would need to look at relay specs) but as you reduce Vin the capacitor charges in a "more leisurely" manner and at 7V the relay may still operate without the capacitor, even while being well below specified voltage, but with the capacitor the slow rise time defeats it.

It's possible that adding a large capacitor from relay center terminal to ground MAY allow lower voltage operation by making a larger source of surge current available.

Here is the relay datasheet all the way from Ningbo (directly south of Shanghai across the very wide river delta).

Guaranteed pull in voltage is >= 9.6V so 7V is "rather low".
Dropout MAY be as low as 0.6V!