Actually I'm sure Rust have changed the situation radically, but in a C-style way. That is: decades from now people would understand that easily, but for many years yet Rust would be something on the fringe of IT world.
C language was invented 50 years ago, in 1972. But DOS was written in assembler and Microsoft Pascal, initially in 1981, MacOS was written in Pascal in 1984, and RiscOS was written in assembler in 1987!
It took almost two decades for the industry to start that switch to C (and by that time C got it's own half-descendant — before it even got it's first standard document).
And then, of course, C# and Java arrived — with tons of hype and promises to make code safer. On top of decades-long hype of pushing GC as the one and only way to safety.
I wouldn't ever understand why industry was so hell-bent on GC, my guess would be that success of AS/400 made designers hopeful that it may be replicated in other areas, but somehow… triumph of managed code never, actually, happened.
No one else was able to create another long-living only-managed-code platform. One of the longest-living was Blackberry OS, I think — it lasted for a decade.
Rust showed the industry that no, you don't need GC for the safety — and this changes everything: suddenly GC is no longer something you have to have but something you may use in certain areas when you want to. Suddenly it's not central feature which underpins all the executions guarantess, but a side-show.
As for “other functional languages”… sure, they contain many interesting properties which are not present in Rust and some which can not even be easily added to Rust… but AFAICS none of these advanced features actually require GC.
I think the true impact of Rust would be seen after appearance of other, higher-level, languages built on top of ownership and borrow model. I don't think GC would ever be added to them, but HKTs and other such things can. And because they wouldn't be GC-based they could easily interoperate with Rust. But I don't expect them to appear any time soon. Rust have to become more popular first before people would develop something like that.
P.S. If you want to see how far from GC-based functional languages things like dependent typing can go… look on Google Wuffs. It's not a general-purpose language and doesn't include many fancy things… yet it solves (and beautifully solves) certain real-world problems.