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Nothing is showing anywhere in cli or logs.

I am able to do an echo test with the webphone successfully, demonstrating that the webphone is connected to the dialler.
However, the webphone will SOMETIMES not accept "call agent webphone" calls to connect to the agent interface.

Use debug mode in the asterisk CLI for core and for SIP. Also check 'sip show peers' in case your sip accounts are unreachable (which blocks any calls to the unreachable devices)