The social problem of the name is less similar to Lisp than I think you're making it, but I do agree it is probably not a terrible thing in the end and could be considered good. I don't think it's going to be impactful enough to "save" or not save the language (as someone mentioned elsewhere, if you called Betamax by a different name, it's unlikely the ultimate outcome would have changed).
The reason I think it's not related to the Lisp issue is because the relation of Perl 5 and Perl 6 was like an unofficial successor with a compatibility mode, like C and C++, whereas the relations between Lisps that create the social problem you highlight of people not knowing about "Real Lisp" are the relations of very incompatible forks with varied functionality running around calling themselves "Lisps" or "members of the Lisp family" and people confusing that with "Lisp". (There were some "Perl 5/6, C/C++" types of relations in the path to Common Lisp like Flavors/CLOS but no one remembers or cares about those outside of Lisp users...) Another reason is the timing, why the name change now, years later -- it turns out Perl (5) is less dead or possessing a stench of death as you put it than people might want, it's still vigorous in its own rights, and so really it makes most sense to just call the new thing by a new name and not destroy both. (Larry's wineskin comment is nice.) The Python 2/3 situation is a closer analog if we counterfactually imagine Python 3000 came out with the current 3.7 feature set, syntax updates, and backwards incompatibility. Python 2 users (me among them) would still have dragged their feet, since Python 2 would still have been useful, still had its own vigor, the same as presently, but it might have resulted in renaming some later version of Python 3 instead of the current situation of renaming the continuation of Python 2 (Tauthon).
Going back to the lisp issue to further elaborate why I think it's not very related to this one, the languages that called themselves Lisp (not a Lisp, but Lisp or Some Lisp) shared a heritage going back to the original McCarthy Lisp. The merely "a lisp" languages used that for marketing, but were actually called something different (Clojure, Scheme, even Racket which has to further distinguish itself from Scheme). You see the heritage in the actual Lisp 1.5 manual, where one of the earliest examples should quickly dispel any illusions today about "Lisp's" supposed purity when they show the function 'length implemented with a program block and goto. By only modifying the top-level function define format and substituting ADD1 with 1+, the program works to this day in Common Lisp. For members of the lisp family that don't call themselves lisp, you're not going to be able to make such a trivial transformation, because there's no shared code heritage, just a vaguely shared s-exp-ness to the syntax. And right after that the manual describes the "pseudo-function" 'trace, which to this day is lacking in supposedly modern languages or requires a bunch of IDE ceremony to set up. It's present in CL and behaves the same, though. Continuing to call CL as simply Lisp seems pretty well-deserved.
The Lisp social problem then is that people run into members of the vaguely defined lisp family like Scheme (especially with SICP formerly being a common gateway for freshmen or HS students) or Clojure and confuse "a lisp" with "Lisp". They can spread their confusion further by releasing a project/blog like "let's write a lisp interpreter!" that can be understood correctly as "let's make a program that interprets code for a lisp¹" but tends to be misunderstood (sometimes even by the author) as "let's make a program that interprets code for Lisp".
Of course I also think this "social problem" is way overstated, especially these days when it's so trivial to dispel the old myths and when the gateways via Clojure or Racket are actually good in their own right and so don't leave the same impressions of "Lisp's a neat toy" that only seeing SICP Scheme could. But if people keep talking about the problem as if it is big, perhaps it will become self-fulfilling, hence my long comment in disagreement. ;)
¹ aka a member of the lisp family based on my vague membership criteria that probably don't even pass Steele's 3-part acceptance test in https://www.dreamsongs.com/Files/HOPL2-Uncut.pdf that requires as its final step (atanh -2) to return a complex number. (Bonus if the correct complex number.)