Understood. However, I believe the default is ON unless turned off by the user.
Shake to Undo is available for many actions other than text fields. For example, it can be used to undo deleting an email in the Mail app. I’m not sure that using it for a totally different behavior (and one that may not be at all obvious that it’s available) would be the best use of the feature.
It’s only three taps to get to the Submit a Log option from a device’s screen. But that said, I see that you are a beta tester. In the beta apps, you can turn on a toggle in Account > App Settings which puts a floating button for log submission 
on every screen. There’s even a new wishlist topic to enable that for the production app (although I would not be in favor of that either).
P.S. Don’t forget to vote for your wishlist topic.