Clearly somewhere in your code you have to explicitly handle tab closing and break the references to allow the GC to do its job. Why not free the resources here while you're at it?
When you say "manual memory management" if you're thinking C-style malloc-free then you have a point, it's very easy to forget a free() somewhere. But any language with destructors can handle these situations without much more overhead than GC-based approaches. Just remove your object from whatever data structure was holding it and let the language automagically run the destructor for you. I find RAII a lot easier to model and think about than "you drop this reference and one day maybe your object is destroyed, but don't really rely on that".
>No one has ever devised a scheme that lets you specify very dynamic lifetimes AND statically checks that no leak can occur.
GC doesn't solve that either. If you're not careful and let a reference to old data dangle somewhere in your code you can effectively "leak" data just fine. When there's no clear ownership and nobody knows who's supposed to clean what it's very easy to end up with data dangling everywhere through complex ownership graphs. GC may mitigate some use-after-free problems (generally by masking the problem) but that is something that can be solved without GC either.
I'm not saying that applications with (justifiably) extremely complicated ownership semantics that also require very high performance don't potentially benefit from using a GC and that it could outperform manual memory management by, for instance, batching the object deletions. What I'm saying is that it's that such applications are few and far in-between, basically 100% of all the code I've ever wrote in my ~15years as a professional software developer didn't have such requirements.
GCs should be an opt-in niche tool used to solve specific problems, not a kludge to let developers write sloppy code and get away with it.