In the cases of America and the British, and Rome and the Greeks, this was explicitly true.
As for Europe, it's not a country and doesn't move as a unit. Different European countries were pioneers in imperialism, others were latecomers, others were mostly shut out and ended up doing more desperate, brutal invasions in less appealing places.
There were so many peaks and valleys of European empires, it's easy to lump them into one. You're right, though——it's not fair to conflate Rome with Byzantium, Hapsburghs, Belgium, Netherlands, France, etc.
As for Europe, it's not a country and doesn't move as a unit. Different European countries were pioneers in imperialism, others were latecomers, others were mostly shut out and ended up doing more desperate, brutal invasions in less appealing places.