Hello
I am not aware of any flag/checkbox that would affect the recording of responses
If you need an industrialised solution, just keep on using the dataverse tables and open a support-ticket to Microsoft
If you are ok with unofficial and unsupported solutions, you may try understand how the browser front-end gets the infos via API calls to the CustomerVoice operational back-end
Just take a look at the browser dev console (F12):
For example, when smone clicks on the name of a respondent, in the respondent list in the report section, to look at the answers of the clicked respondent, the following url is called:
customervoice.microsoft.com/.../getresponseinfo
It's a lot of effort ... but if you manage to reverse engeneering all the handshake and parameters to get the api call to work, you will for sure get all the responses as the API gets the data from the operational database (I am 100% sure about that : I managed to get infos that are not available in the dataverse, the translations, via some other API calls reverse engineered in this way)
In the community last week, there was a couples of guys that were pretty close to find a way to use the API, including authentication
Good Luck