I think it becomes intrusive into the storytelling when it becomes a constant chain of "and then yet another, even crazier thing happened to this one character". Especially when the character started as an everyman.
It's a problem many book series have - one character keeps being the centre of the universe and surviving insurmountable odds repeatedly to the point of incredulity and usually the ante keeps ratcheting up as well so your original ordinary-human character ends up with some kind of godlike powers.
To me, Harry Potter is actually fairly good in this regard, as the character development is at least not utterly unlike 11-18 year olds at school during the re-outbreak of a traumatic civil war, and from the start, there's a clear, in-universe reason why the main character is always the one near the action in the depicted timeframe.
Star Wars did this quite well originally ("why is this rando farmhand at the centre, oh, cool, that's why"). The prequels did a little worse ("here's how the farmhand's father got to the centre, ok well maybe it's awfully convenient the ship broke down right there and a simple blood test reveals The One, but ok") and the recent films rather worse ("these people are at the centre...how? Why? Genetic predestination?"). Notably, the less you explain the Force, the easier it is to keep the story believable and consistent. Again, Harry Potter's vagueness over the specifics of magic helps it.
Something that seems to be getting more common, though it's not new, as series lengthen in real-world time is to do a Star Trek: TNG and start an in-universe subseries where you can have things happen to someone else entirely for a change, but as part of the overall universe's story. It can be tricky if the universe is very predicated on one person, like Harry Potter is. Star Wars has not done well at this, but have tried. I think they keep trying to bootstrap interest from existing fans by character reuse, but the better series like Andor pick a whole new story to tell.
Iain M. Banks was especially good at this and didn't even really have a main narrative, as he'd hop all over the Culture in time and space so you could have the cool stuff happen that he wanted to show the reader, but didn't need to keep powering-up one person (or ship) endlessly. But space opera has the advantage that you can just go and be somewhere and somewhen else very easily. It's a bit harder if you need the subseries to be nearby without constantly dragging the other main characters in, especially when they're further along their upgrade path as you need to explain why they don't just pick up the phone, pop round and stomp the "mortal threat" flat and it's another kind of deus ex machina. Doubly so if the threat is very obvious and world-ending, because surely they'd turn up to that!