Summary: The world looks normal despite Quantum Immortality, because the most likely Universe in which you're immortal is one where your immortality doesn't require much luck.
Recap of Quantum Immortality: The Many-Worlds Interpretation of Quantum Mechanics states that every time some event is observed (e.g. a coin flip), the Universe splits into separate universes, one for each possible outcome (e.g. Heads universe and Tails universe), and the conscious observer will find themselves in one of these universes (e.g. either see the coin come up Heads or see it come up Tails) with a likelihood proportional to the number of ways this event can happen (e.g. you'll probably never see a coin land on edge). Quantum Immortality states that my consciousness will follow the path of branchings in which it survives (e.g. if I set up a highly reliable system that will kill me if the coin ever comes up tails, I can flip and flip the coin and I will only ever see it come up heads).
New idea: Let's take a timeless perspective, and ask "what entire life path am I most likely to observe, from the set of all possible life paths?" Quantum Immortality restricts me to observing life paths in which I do not die. Some life paths are more probable than others. It's not clear what this probability is, but simpler life paths seem more likely than others (e.g. I'm more likely to observe a life path that can be described with any of many sets of physical laws and initial conditions, and less likely to observe a life path that requires a unique, complicated description.) I'm most likely to observe a world that follows physical laws all the time, in which many different sets of simple physical laws could create the world I see. I'm less likely to see a world in which I get improbably lucky.
Thus, I expect to live forever but I don't expect to get lucky.
2nd new idea: Let's generalize this timeless perspective across multiple individuals and lifetimes. Zooming out again and looking at the set of all possible evolutionary paths for a lifeform: it's most likely that I'm one of a lifeform that exponentially grows about as rapidly as possible. Over a long time horizon, the most numerous lifeform is the one that has the highest rate of growth. Thus, because I am conscious and Quantum Immortality appears to be correct, there's a good chance that I will spawn huge numbers of conscious immortal beings during my infinitely-long life.