I did the exact same thing a few months ago but not from ORD, I went from JFK.

My AA flight from JFK had a strong tailwind and arrived 40min early in SFO. This gave me a 4 hr layover.

The Check in agent in JFK gave me my boarding pass for the CX flight. When I arrived in SFO I had to go to a different terminal. The TSA agent there told me I could not enter the gate area as my boarding pass was not valid.

I returned to AA's terminal. They told me my boarding pass was valid, but issued me another one just incase. The TSA agent wouldn't take that one either. The CX check in counter wasn't due to open for another 30 min.

I waithed at the CX counter First class checkin line. A CX agent arriving to work saw me and asked if I needed help. I expalined I just needed a new boarding pass. She printed one for me. She hadn't even removed her overcoat. That is just an example of the service CX provides A rep from a US airline wouldn't have done that. I proceeded to the gate area entrance as I was looking forward to relaxing in the CX lounge.

That was when I discovered the CX lounge opens 2 hrs before the drparture time of my flight. I had about 1 hr to wait.

The moral is, yes you will need to get a CX boarding pass and depending on how long your layover is you may have to wait sometime for it. The CX lounge will be closed until 2 hrs before your departure time.