Humans migrated by their own means, though. Grey squirrels did not get to Britain by their own means, and cane toads did not get to Australia by their own means. Humans transported them, with disastrous consequences for native wildlife. The brown trout has been transported to places it would not have reached naturally and is now a foreign introduction. It can expand its new ranges further by its wonderful ability to smolt and run to sea and use the sea to colonise new rivers... which it has done.
I'm not advocating eradicating it where it occurs outside its native range. Just saying we don't think of it as an invasive species - which it definitely is in some places - thanks to man. It's something to think about...
The eradication debate is very topical in South Africa where trout have been present for 140 years. Some wish to eradicate them completely and some rivers have been poisoned with rotenone. There is a hiatus in these developments at present, I believe, whilst they consider the economic impact of trout tourism and associated employment, but the threat has not gone away.
My view is that after 140 years, any alleged damage has long been done, and they are now part of the furniture. They are contained by climate factors and there is no risk of them colonising where they haven't before (unless there are dramatic climactic shifts).
Some fish rely on other organisms like birds to distribute their eggs in different water courses. Many plants and other species do too. Who is to say humans aren't intended to serve a similar function in the ecosystem? Ok, it is a bit far-fetched, I know, because some of the results have been disastrous, but many species which we take for granted (including cows, horses, pigs, goats, sheep, dogs, cats) wouldn't exist if we didn't have an inherent instinct to mould the environment. Why is it deemed any more acceptable to introduce pets and other domestic animals, which can have detrimental effects (like farting bovines, or the spread of rabies), but not for sport?