Thanks @NanaDei for the quick response. I have followed the browser based SSO connection instructions and ensured the advanced options read authenticator='externalbrowser'. I am still facing some errors however, namely "Underlying DBMS error [Snowflake] ... CURLerror (curl_easy_perform() failed) - code=6 msg='Couldn't resolve host name'."
This (I believe) narrows it down to the server url, which in the Windows ODBC administrator I'm only successful connecting when dropping the https:// from my snowflake server url. Contrastingly in ArcGIS Pro's database connection, I get an immediate error without the https:// and when I add it in, it spends a minute trying to connect before the above error (which is the furthest along the process I have got). This feels like they need to be aligned but I am slightly lost as to why this is happening.
Further to the above, would you mind clarifying whether or not I need to insert my username and password on the Database Connection page (in Pro), or leave it blank and await the browser based SSO to use my details? I presumed leave it blank but without any information, the 'OK' button is greyed out so I cannot progress.