Managing Steam game saves in gaming arcade environment
Solution 1:
What system do you have to identify the user?
You only mentioned not using Steam login, but how will you do the authentication? How does the computer know "Bob" is playing the game instead of "Bill"?
Whatever authentication method you're using, the save file will be managed within that system.
Or the system lets them insert a USB key with their save files on it, and the system copies it over to the right folders, then deletes it when the session ends.