Why isn't it possible to recover/resume after an Exception?
"Resumable exceptions" are indeed a well-known idea in some programming language circles. In particular, Common Lisp has had resumable exceptions for a long time (so they are really not confined to research language; Common Lisp could be considered "mainstream", it has been widely available and relatively widely used for a long time).

For a discussion of resumable (or "restartable") exceptions, see for example this blog post by Manuel Simoni, which links to this dylan mailing-list post by Chris Double and the following Lambda-the-Ultimate discussion: Common Lisp exception handling.

You made the insightful guess, that resumable exceptions are not a good idea in all contexts. They can make resource handling harder and more generally anything related to side-effects must be considered very carefully when you add new control-flow entry points to a block of code.

It is possible for the exception raising code to decide on whether it should be resumable or not. This is natural if you see resumable exceptions as a combination of usual exceptions and continuations: you raise an exception and, in the attached data, decide whether or not to include a continuation to the rest of the computation; the handler then decide whether or not to invoke that continuation, that is "resume" the exception.

Effect handlers (try..catch block, using-like resource handler assuming lexical lifetime) can get confused if you re-enter the computation. Combining handlers and rich control flow is difficult and gave way to a lot of different approaches in the Lisp/Scheme communities (keywords: dynamic-wind, unwind-protect). You are certainly adding non-trivial complexity to the language.

I speak of "effect handlers" because, once you have such rich forms of control flow, you get very close to the ability of presenting arbitrary effects (exceptions, backtracking, but also logging, mutable state and input/output) on top of it. See for example Andrej Bauer and Matija Pretnar's work on Eff a language based on such "effect handlers".