Yes, there is definitely a point to carrying the Admirals Club card, even considering that the membership is linked to your AAdvantage account, for many of the reasons listed above.

Will Qantas Clubs also call to check if you don't bring your Admirals Club card? I remember some Qantas Clubs being unwilling or unable to confirm Admirals Club membership with a phone call, meaning that you had to have the physical card. Maybe they can now.

And yes it's faster and easier to access AS clubs with the physical card, especially if as aisleorwindow indicated above there is any difficulty in the phone process.