Why are sequences of random variables, instead of the sequential observed values of a single random variable, the objects of study in the topic of convergence in probability?
Extrapolating from this example, would another example be a stock's price, where the sequential observed values of a single random variable represent the evolving stock price up to a certain number of observations, and each random variable in the sequence appends a new observed value? The problem with this is the video seems to suggest that the sequence of random variables should represent a sequence of moving averages, whereas my assignments would require taking each random variable's expected value as a further step (which doesn't happen in the convergence formula, and it sounds in the video like each random variable just is an expected value).
What is the correct way to build this model?
Stock price would be better modeled by a stochastic process. In simple terms, a stochastic process is a random element from the probability space $$(\Omega, \mathcal{F}, P)$$ to a subspace of functions, just like a random variable maps from from the probability space to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.
See: http://katzman.staff.shef.ac.uk/MAS362/BChapter5Slides%20PRINTABLE.pdf