There's not as much crossover as you might think. In North America the native pollinators are adapted to the native plants and can't even pollinate the introduced eurasian ones. And it goes the other way: honeybees can't pollinate the native plants, only the introduced eurasian ones.
If course, if you're in Europe, honeybees are the native pollinators. At least around the Mediterranean.
So, what you're saying is that honeybees are facilitating the spread of invasive plants in North America? Seems like a definite negative.
We also have many wild bees in Europe.
While I haven't done an intense study of it, I very frequently see multiple bee species, natives and honey bees, on the same flowers. This includes things like raspberries, mint, dandelions, various fruit blossoms, as well as vegetables. I'm sure there is specialization in at least some of the natives, but some of them, the bumblebees especially (or maybe that's just because they are bigger and easier to see), seem to be pretty generalist foragers much like the honey bees