When I create a new data input (TCP port), where are these settings stored? I would have assumed it would be inputs.conf, but it is not located there.
It should be (see Get data from TCP and UDP ports in the Getting Data In manual).
Are you looking at the right inputs.conf file? See Configuration file directories in the Admin Manual if you aren't familiar with the multiple versions of configuration files and where they sit in your installation.
Usually in the local directory of the app you were in when you created the input.
Or maybe system local
Another tip is using btool to find where it is:
/opt/splunk/bin/splunk btool inputs list --debug