By having it's own templating language, Svelte is able to compile the templates to JavaScript in a way that addresses many of those concerns in a way that I think it easier to deal with as an end user of the framework.
If Svelte were using JSX, I don't know that it'd be possible to do everything else that it does so simply because you can write anything that's valid JavaScript in JSX whereas Svelte has just a couple of basic control structures that its compiler can parse and convert to runnable code.