However I'm not sure If I want to update my firmware because I can't remember where my recovery phrase is. And the software says that I should have it within reach, as a precaution.
Regardless of what you do with updating the firmware... if you can't remember where your recovery phrase is, or you have lost it, then I would recommend that you get your assets moved to a new wallet as soon as practicable. Then wipe the device, generate a new recovery phrase and store it safely, then move the coins to the new wallet on the device.
That 24 word recovery phrase IS your wallet, without that, your coins are going to be lost should anything happen to your device!