Does the test button turn red (I still have no idea why red means ready and green means it's still charging)? Does it fire a test flash when you push it? In answer to your question, no, the flash shouldn't be burned out yet, not even close. Though, if it doesn't fire a test flash then something is wrong. I know there have been reports of some issues with rechargeables. So much so that Canon actually contraindicated them in the labeling. It's nonsense, but still, try fresh alkalines if the test flash isn't working just to be sure.
07/31/2023: New firmware updates are available.
05/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
04/20/2023: New firmware version 1.4.1 is available for EOS R3
03/30/2023: New firmware updates are available.
2/07/2023: New product announcements!EOS R8 EOS R50RF-S55-210mm F5-7.1 IS STMRF24-50mm F4.5-6.3 IS STMRF15-30mm F4.5-6.3 IS STM