Just to straighten out a few things. ( As You keep throwing around your peculiar associations) Thou in the 2 above statements you are clear and straight forward 
HA-Green works great without a Battery ! , whether it’s first time it’s started(years after it left the production-line)
The Battery is only for “keeping” a synced Time(shortly !) ( which it BTW can’t, during the years it lay on the shelf’s waiting to be unpacked and plugged in )
Any OS/NIC will request an IP( If not set to Static on the NIC(OS level) , and immediately upon startup, request a time-sync over NTP protocol from a Public NTP-Server(Or equivalent updated source)
Do the factories “sync” all devices before the leave the “Factory/Store” ? , doubtful
Have you Ever setting up an old PC ( Im sure you have
, Yes the Time is Not synced ! , If it has been in the closet for 1-2 decades
( Battery is dead since long, And Time is Not been “synced” with “the outer world” either ! , for years or decades )( Regardless whether the battery still has a few v/amps left, or still fully “loaded”
However the PC works regardless ( Even i.e Windows OS , parts of)( As long as you don’t try to connect it to the outer world ) NO Battery needed
And the Hardware relies only on the Hardware Clock (ticking, and “counting time”, to adjust/calculating i.e frequencies etc) NO Battery needed
HAOS, startup process:
Check Hardware, found NIC, trying to connect ( For IP and NTP )
One of the very first things it does, with in a few second/minutes
What could possible go wrong ?
Sure COX seems to SucX , the Router seems lacking of this NTP-Service settings, So the initial Routers-DHCP gives HA an IP, and The Router have to rely on COX Internal Network to receive/request an updated NTP-Timestamp
Now HA cant request an external NTP, before it gets an IP ( No devices can’t )
HA relies on firstly getting this NTP, From The Routers(DHCP), as a fallback i believe it’s Cloudflare
How ever & What ever, IF some people doesn’t have enough experience or understanding of their Hardware and Network, there could be so many “pitfalls” In the Process of setting up an Device(which rely upon NTP, like an OS) ( With both an Hardware-Clock & A Software-Clock )
I.e
How many read the install/setting-up/get started carefully before the plug it in i.e their HA-Green ?
( So they basically fire it up/plug it in to see whether it lights up/start spinning/what ever ) … Before they even think of their next move
( i.e plug in a TP-Cable, and make sure their Device has Network -connection (nor confused with internet) , HA-Green ( or Hardly any Hardware/Software “Waits” for this, It just states " Ok, this didn’t work, … waiting 90seconds,move on " ( Software Clock tick on "watch the install process
)
No Many people simply don’t take “precautions” do to lack of common Network/Hardware and Services experience
Op is not in the best situation, with his COX Hardware, and best would be to buy/install a Better Router, skip their COX “Modem”
However this will not change the fact that HA-Green will work without a battery, but it wont be able to connect to other Device on the internet, which rely and depends upon correct NTP-Time ( No Devices/OS will )
Even if they have a battery they will still need the correct Time, in basically seconds, for every i.e HTTP(s)-request
I Still Doubt it was the “Battery installment” which fixed OP’s situation, most likely it was just the fact the he remove/shut down his Green, and let it “rest for while” dropping out of his Router
And when he tried again ( with battery ) , he “Unintentionally” followed the guideline in a more “structured” approach
Im sure the Battery he installed didn’t has an NTP-Updated time stamp, so in fact it was the initial NTP request which failed, most likely do to the COX Hardware/Software, or inappropriate/faulty install/setup procedure in the first place
Just to make it clear, Next time you restart your Device, the OS/Nic will request an NTP-Time, It’s not coming from the “Clock” which the Battery stores the latest it “Had” , it’s Updating it ( NO Battery needed, again )
The specific requirements or Rules/Regulations for a Device-Clock is pretty clear, but having a battery to make the “Shutdown” Hardware “Clock” ticking , without getting regularly updated for years is Absolute Not 1 of those Rules/Regulations Nor Requirements, If it’s a Device( Function/Feature ) which rely Upon NTP
Battery or Not, You need the NTP-Service to work properly in your network