Sure, it's notionally possible, but churches provide structure and organization for it. For every one secular food pantry there are ten plus run by a church. And if you shut the churches down, it's not like there are suddenly going to pop up dozens of secular food pantries in their place.

Church attendance and giving dwarf any other form of charitable donations, and is therefore a hard thing to replace. Most people I know who go to Church regularly are giving 10+% of their income to the church. Of my friends who don't go to church, I don't think I know a single one who's charitable donations for the year top $5k, and for most it's sub $1k. And I live in a pretty high income area.

It turns out that having mandated charitable giving leads to higher rates of charitable giving, and this gives these institutions the bankroll necessary to do things like run soup kitchens and shelters.

The only other source of money for this is the state, which notoriously underfunds and mismanages this type of work, so churches are filling that gap. So the best case scenario, if all the churches close, would be if the state raised everyone's taxes to get extra funds to offset the loss of that charity. But it turns out that that's not a politically savvy move. People hate raised taxes, and most hate the poor. And poor people in general don't vote in numbers big enough to help you win, so you're not even buying political good will there. So if the churches go away, it's not like there's a real backstop that can come fill the gap.