I suppose it's time to be precise ...

The model-specific manual that the manufacturer provided says:
"Grounding the Generator: The National Electric Code (NEC), as well as many local electrical codes, require the generator to be connected to an earth ground before operating. Before starting the generator, make sure it is connected to earth ground by connecting the ground terminal on the control panel (See Figure 12) to earth ground using copper wire (minimum 10AWG). Consult a qualified electrician for local grounding requirements."

(OK, so my earlier statement about #8 wire was incorrect).

Now, what else is in the manual?

Well, I see that they describe the wheels as an "accessory." I also see a section about connecting the generator to a transfer switch.

Let's pause and think about that for a moment. Is it possible that some variants of this generator are permanently installed?

I think we may have found the electrician's equivalent to "one size fits all" clothing.

I see absolutely no benefit to bonding this generator to "Mother Earth." I do see some (marginal) benefit to having a redundant bond to the equipment grounding network of the building.

Using the generator as I intend .... physically unplugging the furnace from its' receptacle on the wall, then powering the furnace with a (3-wire) cord that runs straight to the generator ....., well, I don't see any way making a 'permanent' ground connection improves things.

Of course, we've had our discussions about having a flexible cord on a permanently installed appliance, such as a furnace. That's another topic. personally, I'd rather see a cord on a furnace than find a 'suicide cord' for the dryer receptacle.

How will I get the cord out to the generator? Well, I'm going to try something new. Garvin makes an "extension ring" that takes screws on BOTH sides. That means I can place a device ring on both sides, making an opening right through the wall. I could mount an "inlet" on one side and a receptacle on the other, or I can simply pass multiple cords out the hole, stuffing it with fiberglass to stop drafts and bugs.

(I plan to use a similar arrangement for my phone / data / CATV services).