I don't see how that would prevent what Germany does. If a State wanted to withhold tithes from people's paychecks in that State, the 1A doesn't prevent it, as long as the government doesn't favor any one religion. As long as any religion could apply for this service, it should be legal. The problem is that it'd probably be an administrative nightmare. As I understand it, over in Germany, most Christians still fall into a handful of denominations, which are all probably organized at the national level (i.e., the Catholics have organizations at the diocesan levels, and probably one country-wide level above those, which reports to the Vatican; the Lutherans have one organization, the Anglicans too, etc.). Over here in the US, things aren't that simple. While the Catholics are of course well-organized, the other mainstream Protestant denomations are less so: there's mainstream groups for the Epsicopals, Lutherans, etc., but all these also have renegade divisions where some chuches at some point rebelled against the heirarchy and split off into their own sect. The Lutherans, for instance, have the Wisconin and Missouri Synods which are ultra-conservative, unlike the regular sect. The Presbyterians have PC-USA which most churches are part of, but a bunch are either independent or part of some other ultra-conservative group (lately in response to the Presbyterians' acceptance of homosexuals and of homosexual preachers even). There's a zillion different Baptist groups out there dating from the 1800s. And these days half the Protestants are Evangelicals, and frequently part of some Prosperity Gospel megachurch, which is totally independent. All in all, there's probably tens if not hundreds of thousands of "organizations" around the nation, just for Christianity, though most of these are independent churches both large and small (some of them in peoples' basements even). So keeping track of all these entities and giving them access to the government-tithe-withholding system would end up costing an absolute fortune. In Germany, they probably don't have this problem because 1) I'm pretty sure they don't have remotely as many independent churches and 2) they don't have our 1A, so they can probably safely ignore smaller religious organizations and just do this for large, established ones.
Honestly, I'm not sure why Germany still does this at all. Much of their population isn't religious any more, and if people want to give money to a church, they can do it themselves without the government's help. It's probably some silly holdover from previous generations when churches were a stronger part of civic life, but for an advanced and secular western nation, it's really an embarrassment IMO.