Linearity of expectation is of course true in general, but in this case summing the expectations of the individual transition probabilities is only valid becaus

Linearity of expectation is of course true in general, but in this case summing the expectations of the individual transition probabilities is only valid because the only valid state transitions are going from state i -> i+1 (owning i unique toys to i+1 unique toys) or just staying in state i. If you could magically go from state i to any other state (you lose unique toys or gain more than one unique toy), then this simple calculation would not be correct.