No they don't. All they have to be is a bonafide "Non Profit Religious Organization" as registered with the state's Secretary of State. It can be a corporation or not. It must meet the requirements, but it most definitely does NOT need to be a 501(c)(3). This has been discussed dozens of times here on FR, and each time I see it I end up telling people this. The IRS AUTOMATICALLY recognizes a church as Tax Exempt, whether it files as a 501(c)(3) or not.
My point is that many churches do become corporations for various reasons and I understand your view that some do not acquire corporate status yet are recognized as a tax exempt organization. In either case the fact is that the IRS holds the 'tax exempt' hammer over the heads of these churches to the degree that if they determine that the pastor spends far too much sermon time talking politics or criticizing public policy of the fedgov then the church may have it's status revoked. That's why most pastors are so milquetoast when it comes to homosexuality, abortion, illegal immigration, earth worship (gaia), the degradation of public education, etc. In effect they become odorless, colorless, tasteless, charlatans.