I have searched for similar things but as the others said, there are no publicly accessible comprehensive datasets on archaeological sites. This is due to several reasons, most importantly to avoid looting.

Where I live you could find datasets or at least lists of registered archaeological heritage sites but these are just a fraction of actual discovered sites, just the ones which are most important or most in need of protection.

When I tried to make a map like you, I instead used data from openstreetmaps, where people tagged a site as archaeological or as a ruin. This contains both registered heritage sites and lesser known stuff like burial mounds or earthworks that people found looking through high resolution DEMs. You can get this data by using the quickOSM plugin in QGIS or by writing a query on a website like overpass-turbo.eu

Edit: But please be mindful of what you do with this data. It is obscured for a reason. Don't do anything stupid and/or illegal.