I agree with you that some churches are better than others, and that the good ones will gladly open their books for you. I don't know that most secular non-profits are more thrifty than most churches however. We can probably agree that public sector organizations have some large inefficiencies. (I used to work for the local County Health Department, and fully know both the good and the bad of a government run health department).
As far as tax-exempt status goes, you do know that churches pay property tax, right? So they don't necessarily get all those local services 100% gratis. They don't pay income tax, so the federal level services *are* free - but now we're talking military protection from invading armies, the EPA, Ag Department, National Parks, banking regulations, and bond debt service, et. cetera. They also don't collect sales tax; so the city (not county) services *are* free. The funding can get a little blurry - the police officer patrolling the street is probably city funded (sales tax), but the officer guarding the prisoner in jail is almost certainly county (property tax). If you live in a state without sales tax, then the church does benefit a little more than those in other states.
I'll also point out that clergy don't pay income tax, but they don't get Social Security either. And only the mega-churches provide retirement plans. I know my little church couldn't dream of it. I don't know if your employer takes out SSI. But if they do, that's a tax funded safety net you have because you pay taxes. Don't assume that those that don't pay taxes get all the benefits you do.
Lastly, there's the difference between non-profits that get to bill the government for services provided, and churches, which don't. Technically, my local hospital is a non-profit, but the amount of money it sucks down is shameful. $60 for a single tablet of Motrin. It's clearly thievery; but, as long as they can spend 9/10ths of their income, they can claim they only cost 10% admin overhead. I think that a church that hosts an AA meeting is a far more benevolent organization than, say, The American Red Cross, who takes donated blankets and sells them to disaster victims. The charity provided by a church is 100% charity. Not so with most secular non-profits.
I can see where you don't trust an organization that doesn't publish it's finances. But I've also taken enough bookkeeping and seen enough financial finagling (in all sorts of organizations) to know that a claimed overhead rate, even when published, isn't a valid measure of trustworthiness. Which is why I suggested that to know the real truth requires actual in-person knowledge. Even if they published their ratio of income/outgo, that isn't as valuable as knowing if the travertine is in fact a gift from a wealthy parishioner. And if it isn't, shame on them.