Purple spots on phone screen

I'm with Brian in saying that a new screen will almost certainly fix the problem with your display. What you're seeing is pressure damage from the components inside the phone behind the screen. It looks to me like the screen fell face down, and the weight of the phone smashed the inside parts against the back side of the screen hard enough to damage it. Compare these pictures to yours and you'll see that you can make out definite features, most notably the battery and the fingerprint sensor.

So yes, order that screen, replace it, and you'll be all set. Here's a direct link to the guide you'll need to do the replacement.

Google Pixel 6a Battery Replacement - iFixit Repair Guide

And you can even buy a genuine Google screen right here as well. If you don't have the tools already, the Fix Kit version is a great deal; for an extra $7 USD the screen comes with everything you need to do the job, including tools and the screen adhesive.

Google Pixel 6a Screen - Genuine

Good luck and let us know how it all turns out!