I think contrary to your assumption, if we taxed churches it wouldn't affect the church financially all that much. What it would do is affect behavior:
- There would be no prohibition against churches taking political stances and being politically involved. The only thing that prevents churches from taking political sides in the modern USA is that they stand to lose their tax exempt status if they do. You would fully expect churches to come out and explicitly approve or disapprove of political candidates and laws. If the church felt threatened then they might even start voting as a bloc in order to protect themselves. Imagine a Billy Graham figure who decides to run for president on the platform that he's a Baptist minister and all 50 million Baptists in the USA automatically vote for him.
It's important to remember that separation of church and state goes both ways.
- There would be no tax incentive to do charitable giving through the church, presumably reducing overall giving and shifting some giving away from religious institutions and onto secular charities. A little more than half of all charitable giving and volunteerism goes to religious institutions (and the highly religious only make up one third of the population). Secular givers make up only a quarter of the population, and only donate about 10% of all charitable giving and 20% of all volunteering.
But realize that the tax benefits for charitable giving are pretty minimal (you'll always still have more money in your pocket if you just keep your money and pay taxes on it) but there will be less incentive. Moreover, we've had the "standard deduction" system for a long time now, which means that there's no incentive to do charitable giving unless you happen to do a lot of it. This means that religious people will probably still keep giving, they'll just give less than they did before.
The other aspect here are the roles of religious versus secular charities. There are many traditionally religious charities that don't have many secular counterparts. For example, in my city there are literally no secular charities that run homeless shelters or food kitchens- which is common in many places due to organizations like the Salvation Army and the Catholic Church taking up those roles in the early 1900's. It's also the case that 100% of the inner city hospitals in my area are religiously affiliated, though they are no longer religious institutions. A reduction in giving to those religious organizations would disproportionately impact some populations that depend on those services.
So what are the financial impacts on churches? If you taxed churches like businesses, then:
The church only pays income tax on any net profit they have at the end of the year. Unlike individuals (where every dollar is taxed) the church can simply avoid paying income tax by making sure that any excess money they do get is spent on charitable giving or church operations.
The church would be subject to other taxes that are harder to avoid, like property tax on church land, church buildings, and church vehicles. They would have to pay sales taxes. They would also have to pay payroll taxes for their employees, meaning their employees would be about 15% more expensive.
What's the effect here? Well, any given church is either going to have that extra money to spend, or they aren't. If they don't have enough revenue to support the new tax burden then they probably have to close. If they do have that extra money, then they spend a little more on taxes, and they spend a little less on the places that churches usually spend their money: their staff, their building, and their charitable mission.
The result here is that some particularly small or otherwise financially unhealthy churches will close, and those people will go to other, larger churches. The larger combined church is healthier and has more assets. But that's already true today- small churches that take in less than (for example) 100,000 a year would have a hard time paying a pastor, paying the utility bills, and keeping the church property maintained. If you taxed churches you might raise that minimum bar to something like 150,000 or 200,000, meaning more small churches would close, but the underlying phenomenon doesn't fundamentally change.