I'm having a weird behavior here and I can't figure out what's going on. I have a ScrolView with a form and some inputs and labels inside and it seems to only want to scroll when you place your finger (or cursor in the demo) over an input or a switch and it doesn't scroll when placing your finger anywhere else in the ScrollView
I put together an Expo Snack to show the code and if you run in the emulator and attempt to scroll the ScrollView by placing the cursor over one of the labels or on the edges of the ScrollView it won't scroll but if you place the cursor over an input or a switch it scrolls just fine
The issue is because Touchable effects in your TouchableWithoutFeedback are blocking the effects of ScrollView, therefore you need to reset your responder by wrapping the Content inside the View
<Content>
<View onStartShouldSetResponder={() => true}>
//... Rest of the code
</View>
</Content>
Also don't use ScrollView since NativeBase uses KeyboardAwareScrollView from the package react-native-keyboard-aware-scroll-view in the <Content/>