Pip probably installed exifread to a different version of Python than the version ArcGIS Pro uses. The Python version that installs with ArcGIS Pro is a walled garden. You cannot install third-party libraries to that version of Python unless they are ESRI authorized and preinstalled with Pro. You can clone your Python environment (create a new version of Python) on your computer and use that cloned environment to install third-party packages.
These ArcGIS Pro Python versions come preinstalled with the conda installer. Best to stick with conda for installing rather than pip to avoid conflicting library versions in any cloned versions you may create.
I see that the default version of Pro Python includes pillow-simd (a fork of Pilllow). Pillow, and PIL in the 2.7 days was capable of reading EXIF data. Maybe investigate the exif reading powers of pillow-simd. Here is a link to get you started. This will save you a lot of hassles having to have IT clone your Python environment and recloning it when they have to reset your roaming Windows profile.
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