In the left panel shows the History tab, which I can view previous requests I've made. When I click any of the previous requests, it displays the URL and key-values used, but doesn't show the response.
Is there a way to view the old response?
Yes, there is a way to show previous responses but not via History tab.
What you should do is:
Update:
In recent updates Postman added possibility to view saved responses from History tab:
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