Can Lightroom apply the same center crop to multiple photos at once?

Yes. Lightroom can do this directly.

Select the photos you want in the filmstrip, then use the Crop & Straighten tool (shortcut: R) on the active image. If the images are selected together, Lightroom can apply that same crop to all of them.

This works well when you want the same centered crop across a large batch. If some photos need the crop moved or adjusted, you’ll need to edit those individually or in smaller groups afterward.

So for your workflow: apply the common crop in Lightroom first, then fine-tune any exceptions, and export from Lightroom when done. No need to send everything to Photoshop just for this task.