I cannot for the life of me get into my netgear APP

Chicken and egg problem. You can't make a new phone trusted by using a 2FA method supposedly by receiving push messages. Allowing this would kill the advantage of 2FA. The push messages would go to your old phone. You can try to bring up your old phone by WiFi only...

Beyond, I'm honestly a little bit lost because of Netgear has removed the ability to change the 2FA SMS number from the Netgear Account Profile Login Settings.

Further on, even if Netgear does allow it: It's a litle bit blue eyed using 2FA based on Push only - which can go to a specific device only. The device can fail, can be lost, ... 

On some Apps like Insight Netgear does offer using email for the 2FA part (beyond of push and SMS) - this might help, permitting not everything is bound to a single email without any other liked accounts where a friend or trustee can help to regain the access in case.

This used to be the workaround for changing the 2FA primary SMS number: ... to my knowledge but there was never one for push-only:
===
Login your MyNetgear account.
Go to Manage Profile.
Login Settings.
Two Step Verification.
Add your new SMS Text number.
Then you delete old and make your new Text primary.
===

This is not available currently why ever.