They are unsafe because they invariably result in people either ignoring them or catching more than they should, or less than they should, and the compiler happily lets you do that, EVEN when you're using checked exceptions.
There is no should/shouldn't. If you don't have a specific error in mind and how to handle it, you shouldn't handle it. In practice most errors can be left unhandled all the way to e.g. server response, so this works quite fine. `try-with-resources` is typically awkwardly implemented unfortunately (defer is nicer, the new `using` keyword in JS is quite nice too)
I don't mean to be uncharitable but it seems like most exception advocates here don't really understand the case us functional error handling advocates are making. We don't advocate handling errors at every level. They're usually sent to the boundary layers of the system, eg the http resource layer, and folded there into http responses. The safety case we're making has nothing to do with try catch finally, it is about signatures and what callers are forced and allowed to do.
