House sparrows follow human agriculture, they're adapted to that.1
They don't do that to conquer but because they must - that's their job, to take their place within human *aggression*. In following human agriculture, they're actually following human aggression. Humans usually (in their hypocricy or in plain cruelty) don't realise how aggressive *their own* way of life actually is.
Thus, if house sparrow is "causing problems" it just takes it's place. It's humanity causing problems, and going to places they don't belong. It's human who have built and utilized bluebird's natural habitats to their needs, or actually to their *greed*.
When nature seems to be causing problems, it's always actually human greed. When nature appears within human greed, it gives a *sign* of it. In christianity, the bible says god counts every (dead) sparrow (Luke 12:6). But like always, it falsifies the issue giving the whole thing as a playground for experience of presumed (individual) human value (Luke 12:7). Truly more deeper meaning the phrase would get if it said: humanity is greedy, and nature sees what is happening.
Nature reacts and gives signs as long as it needs.
About invasive species people may say they are "disheartened" of what people have *done*, but I'm disheartened of what people *do*. There's a centuries of difference.
People naiively think that watching birds nesting can be self-evidently a pleasure. There's may be a lot of crows, gulls and other aggression and unfortune which can take many forms, and which self-assertive human presence very easily increases.
People are speaking of "our native species" and "beautiful country", but ownership and experience of beauty all contain violence based on the accepted way of life. People don't know what drives nature, they think they own it. Biologist in general don't. They're introducing their view of doing violence: trapping, testing, killing - and justified violence in nature is exactly to be directed by belonging to "invasive species". Animals see what happens around us, even when we don't, and they react in ways that can (in turn) "dishearten" us, if we don't already be awared of that. We might not be entitled to joy, let alone ownership. See also https://socratic.org/questions/how-did-native-americans-and-europeans-views-on-land-ownership-differ
Ornithologists show pictures of hawks, pictures of hawks with prey, pictures of hawks with prey still alive. We are far from any solution. People are only inventing new ways of using aggression and getting joy of it.
(Extinctions and isolated island are completely different thing, and even then the practices can vary. I'm talking about backyard esthetics and such words as "historic highs" and "pleasure", whole way of life based on unnecessary killing.)
1 When human have done their conquests, different animals may have taught them to do that. And they may be the exact one they direct they're anger to. Thus the relationship has existed before human did what they did. In that case human way of life as such is dependent on that relationship. If the animal hasn't been capable to follow the next conquests human has brought them with them one way or another.