The creative cloud app needs to at least once a month see the internet to get new authentication keys. So make sure you leave it running and updated. If you are still on Lightroom CC you are running a very old version so make sure you keep Lightroom Classic updated.
Are you saying that sometimes Lightroom (it's no longer called "Lightroom CC") does work as expected without an internet connection, and was it only the last time you tried it wouldn't start? If so, did it produce any error message?
Generally, I find it works OK most of the time, though I have seen issues when the connection from my computer to the LAN is still working but the connection from router to the WAN is down. If I force a "not connected" state by disconnecting my computer from the LAN, starting Lightroom in that state works fine.
The Adobe Creative Cloud Desktop App ACCDA) is used, amongst other purposes to:
Install the various programs like PS, Lr, LrC, etc
Uninstall the programs
Update/Upgrade the programs
Install Adobe sourced Plug-ins, etc
Authenticate your subscription
Activate the programs
Deactivate the programs
A key part if that that relates to your problem is Activate/Deactivate. When you sign in via ACCDA it checks your subscription, authenticates it,and activates the Adobe programs on your computer. When you sign out, it deactivates the Adobe programs. You need the internet to sign in. You do not need the internet to sign out. As long as you do not sign out, internet or no internet, the Adobe programs will not get deactivated. Do not sign out.