I don't think the problem lies in GSAP or staggered animations.
When you click you are using SplitText to break that text into characters and words and then inject 100+ new DOM nodes (with their own inline styles) into the DOM. In addition, there is also logic that needs to run to preserve the styling of nested tags like <span>, <strong> etc.
All that has to happen before any animation comes into play and THEN the staggered animation needs to create 100 or so more from() tweens, which involves assessing and recording start and end values.
And to make matters a bit worse, by using 3D transforms, each of those 100 or so nodes need to be layerized / sent to the gpu.
I suspect that's a lot of work for browsers to do instantly. I'm not quite sure why Safari takes more time than other browsers, but I don't think this is a problem inherent to staggered animations. My guess is Its more related to using SplitText, the amount of text you are splitting and the amount of animations you are creating.
My advice would be to split the text before you need to animate it if at all possible.