Yes, you are theoretically correct. It is as you know the other way around out west where they consider the Brook Trout to be a nuisance. I also believe that the park manager do actively manage the brook trout as for reintroduction and protection versus the non natives, brown, rainbow. I guess we are lucky to have the non natives. If we really look at the mandate for the park. To protect and conserve native flora and fauna. The rainbows and browns are really an invasive species here in the park. It's a complicated issue but I have many times thought along your line of thinking. I have more than once reintroduced a brookie in to the pool below me and taking out the rainbow and putting that fish even lower.
I doubt it has any effect but it's fun to do anyway.