Hey there m_a_thrasher,

We understand the option to allow a sign in through two-factor authentication is disappearing quicker than expected. Also, the Allow button does not always respond. We'll be happy to help.

The prompt to Allor or Don't Allow a sign in should stay up until you dismiss it. With that in mind, if you share an Apple ID, it's possible the prompt is being dismissed from another device.

If you're unsure where all your device might be signed in or who has access, you can change your password: Change your Apple ID password

We'd also recommend making sure all of your devices are up-to-date.

Additionally, as you mentioned, there are other options for getting a two-factor code. Reference this page: Get a verification code and sign in with two-factor authentication

If you still experience issues with this, reach out to Apple Support directly, here: Get Support