Q. Is it possible to get a start time of a call using jtapi?
(I understand that adding an observer, capturing call connected event, and then recording the time is one way, but is there a stored property somewhere where this is tracked?)
A. You can probably look into the CDR interfaces if real time info is not needed. Search for Cisco SOAP - CDRonDemand Service on Devnet for API to interface with CDR.
I don't think jtapi has any such stored item. Like you said, I think the only way is observing, which you are probably doing anyway if you are using jtapi. First events are typically CallActiveEvent and/or ConnCreatedEvent.