I think you're mostly correct about churches focusing on wealthier people because they need the money, but there may be a feedback loop in play too.

In much of the United States, active church membership and the networking it offers are necessary for community leadership positions and success in local politics. Voters are traditionally reluctant to vote for non- Christians, though that phenomenon seems to be fading some. More than that, membership in a respected church conveys an imprimatur of respectability.

All that to say that I think sometimes wealthier people are motivated to be active church members because doing so helps fulfill their community leadership and political ambitions.