I'm inclined to try to learn a Lisp, but the tight integration with Emacs is putting me off. I spent a lot of time and effort to learn vim and I'm wondering why a family of languages should be this dependent on a specific editor...
As mentioned by others, its not. But Emacs is also a Lisp runtime. People who are attracted to Emacs eventually learn some Lisp (Emacs Lisp, closer to CL than Racket et al) or were attracted because of Lisp in the first place. As a result, the overlap between CL devs and Emacs users is naturally higher than that for other editors. This translates into people developing more and better tools for their editor, and higher popularity usually translated into more and better tools (with obvious exceptions).
As mentioned by other commenters, SLIME can be used with Vim too, if that's your thing. But just like SBCL, Emacs is kinda of de-facto editor of choice, not by force, but by love of its users :)
I used CLISP because it had GNU readline. (e) to edit my file, (l) to reload it, and off to the races (I wrote those trivial defuns). No packages, no imports, no asdf. Everything in CL-USER.
I did switch to emacs for one simple reason: tab to reindent. Other than that it was the most basic of emacs use (load file, save file, ^s to search, cmd-XCV for cut and paste, pgup, pgdn). No Sexpr navigation, no buffer juggling, none of that, no lisp integration whatsoever outside of paren matching and the tab to indent.
The language isn't, but SLIME is because emacs isn't just an editor, it's a platform and system that people develop applications on top of. And the community was already there, using emacs, to continue with that option.
It just happens that SLIME is a pretty well designed UI (the acronym stands for Superior Lisp Interaction Mode for Emacs, so I guess that was the goal). It’s miles ahead of what you get from most other languages, even C++, but most of what makes it so great comes from features that were already present in the language implementations. It also combines very well with other advanced Emacs packages such as Paredit.
Interestingly, SLIME is implemented as a network protocol to allow for remote debugging, so you can implement a new UI for it any way you want. I’ve never used it, but you might check out SLIMV instead. https://www.vim.org/scripts/script.php?script_id=2531