The EWS reporting features are an As-Is service. You cannot customize them any more than how they come out of the box. If you really want to expose more details about the printer through automated reports then scripting is your only solution.
Yes, you would have to designate a machine to host the script and run it at regular intervals if you decide to go down that path. This will depend on the expectations for your reporting. If they are only for youself then you can set up a scheduled task (windows machines) to run the script at various intervals and under certain conditions. Obviously once you leave the network or power down the reporting will stop. This is a problem if you have a backup or someone else who needs to fill in for you while you are away.
The solution from there would be to designate a real server to run the script for you. Something that is powered on and connected to the network 24/7. Normally a good candidate for this is the Print Server itself when one is being used. Since the print server is normally running at the same condition as the printers it supports its an easy choice to keep these tasks bundled together. However, any server on the network that you have access to would do the trick.
If scripting isnt your thing you can always contract a little project like this out to someone. I'd think a high schooler looking for referances or IT intern would be a good candidate to put it together for you. Let them do the foot work to get the script going and then take it over for general maintenance from there.
Happy to answer any further questions you have.