Hi James,
Thanks for your post in this community.
Based on my test, if the site is created in classic experience, then I will have this Excel web part when creating a classic page.
However, if the site itself was created in the modern experience, then the classic page you created will only have limited web parts. As shown in the image below:
Hence, please first check if the site you are using is a modern site or classic one. If it's a modern site, then I would recommend using the File viewer webpart in the Modern pages instead of the Excel Web access webpart.
Meanwhile, I would like to confirm if you are using the SharePoint Online or SharePoint On-Prem, the test I did is in SharePoint Online, so the result might not be applied to the On-Prem environment.
Please keep us informed if you need further assistance on this issue.
Regards,
Alex Chen