Your mention of native trees that can also be problematic reminds me of one tree that is a pain in the &*&*&. Our native oak trees. Tall, great shade trees, valuable wood. However... their acorns are beyond problematic! They take root wherever they end up and grow deep main roots very quickly. Just a couples years old and they're hard to pull up. I had spent countless days pulling up these darn trees that were growing everywhere - even in flower pots no where near the tree. The squirrels don't seem to even like them much anymore. Plus, the acorns fall hard on whatever is near them, like your cars. even your head! I had my husband cut down every oak tree that was in the vicinity of the yard area of our property. Fine if they're in the woods. Seems like just about anything becomes invasive in it's own way.
What I don't understand is why there are local nurseries as well as numerous online sites where the non native plants and seeds are allowed to be sold.