It might’ve been a factor, but it would probably have been a combination of factors, like being farther away from the hub, using it a lot, and may be a partially depleted battery when you got it. It’s Spec’d for over 3,000 presses, so you didn’t do that just from setting it up. 
One a more detailed level, I am interested in your comments regarding pairing the device with the hub. For example, if the hub is in the living room, then one’s subsequent use of the Fob would ideally be at points close to the hub. I mean if I want to have the Fob on my keyring (like your cool photo above), and use it to arm/disarm my system (just like my car fob for the car) then I should do that as close as possible to the hub. Is that correct? That simply comes down to a behavioral issue.
Yes, that’s correct. That’s why the user manual for the device says the battery specification is based on a “direct range.“ Meaning a direct connection to the hub.
As a side question, does the Fob have to be ‘woken up’ or does it simply normally operate like my remote car fob/lock ?
This one is more complicated, it’s not working exactly like your car Key fob probably works.
There are several different factors involved, depending on how you set it up.
Normally, any button press would result in that scene code being sent to the hub. The device will sleep until the next button press.
However, the device does allow for a locking code to be implemented. If you use that, it goes to sleep and stays asleep until you press the unlock sequence.
Additionally, if you set up a multi button sequence to activate a scene, including double press, then it wakes up as soon as you press a button, but it waits a little bit to see if you are doing the first button in a sequence. People often misinterpret that as the first press being ignored, but it isn’t. It’s just how they are handling multi button instructions.(the other way some devices do it is by having a mode button that you press to say that you are then going to enter a multi button sequence but that’s not how this device did it.)
And the multi button sequence rules also apply to double click, and triple click.
Activating a double click will introduce delay to a single click reaction and activating a triple click will introduce delay to a double click reaction.
So again, people sometimes interpret that delay is meaning that the device is asleep, or not working when it is, there’s just this introduced delay to make sure all the buttons are captured before the action occurs.
And finally, although this device does support zwave direct association, it’s not actually very good at it, and it gets pretty easily overwhelmed if you have a lot of devices in the association group.
It is not recommended to associate more than 10 devices in general, as the response time to control commands depends on the number of associated devices. In extreme cases, system response may be delayed.
So all in all, it’s a somewhat quirky device and you have to get used to its response pattern. It’s definitely not the same as a regular car key fob. 