Here is a rephrasing which simplifies the intuition of this nice puzzle.
Suppose whenever someone finds their seat taken, they politely evict the squatter and take their seat. In this case, the first passenger (Alice, who lost her boarding pass) keeps getting evicted (and choosing a new random seat) until, by the time everyone else has boarded, they haveshe has been forced by a process of elimination into theirher correct seat.
This process is the same as the original process except for the identities of the people in the seats, so the probability of the last boarder finding their seat occupied is the same.
When the last boarder boards, the first boarderAlice is either in theirher own seat or in the last boarder's seat, which have both looked exactly the same (i.e. empty) to the first boarderher up to now, so there is no way the poor first boarderAlice could be more likely to choose one than the other.