You absolutely can use Google imagery. Here's the pricing page [0]. I also don't see how other imagery providers wouldn't be similarly effected, unless the issue is specifically with Google publicly distributing this imagery for free.
There are also much more relevant green space vector datasets that you should be using before using aerial imagery and object classification for any of the tasks you mentioned.
It's quite common to use aerial imagery in archaeological remote sensing.
[0] https://developers.google.com/maps/billing-and-pricing/prici...