Sure, you can check for everything. It often is more effective, but not prettier or easier to read and of course you also will miss cases.
In that case a message from a network either sends or fails and in the case of sending try/catch does nothing for you but provide an error object, and you don't need try/catch to capture the error. In the case of receiving a network message the message comes in or it doesn't. If the message does come in and is malformed your application should be mature enough to handle that appropriately to the benefit of the user instead of crapping out some error messaging to the console. You don't need try/catch to do any of this. A simple if condition is more than sufficient.
> check for everything
You should check for everything. You only need to check for one thing and if you check for it you are already at 100% of everything. Did you get what you expected: Yes/No?
Usually you don't need to just know if there is an error or not -- sometimes you need to know what that error is. For example, an I/O error versus your image-is-too-small error -- you need to respond to the user differently. I've seen plenty of web apps that treat I/O errors and user input errors as a generic "an error happened" and that is completely wrong. Proper exception bubbling means none of that code that encounters the errors needs to necessarily know how to handle the error and your app always responds correctly.
That said, I don't use exceptions as much in JS because exception handling in JS is still very nascent. There's no base library of exceptions and telling different exceptions apart is not built-in to the language. In a language with more mature exception handling, it's a breeze.
Its an equivalent breeze in JavaScript as well unless you are fumbling through abstraction layers (frameworks) that eliminate the information you need.
Since you are saying that you have to describe the problem within the error object, it sounds like you must be you are parsing error messages and stack traces to figure out what the error is. That's not how exceptions are to be used and I think that's why you are not seeing why you would bubble errors.
In other languages, you don't have to do any of that nonsense because object identity is much stronger, but JavaScript uses prototypical inheritance so you can't really tell if an error is an ImageError or an IOError. Despite this issue, exceptions were added to the JavaScript language without resolving that problem. The reason why you have to worry about frameworks eliminating error information is because that problem wasn't resolved and so frameworks roll their own error handling and there is no standard.