My question should have been more precise. I'm trying to decide how hard to push my experiments. So, is $p$-adic continuity ($p = 2, 3$, or $5$) of $n \mapsto x_n$, or the lack of it, already a theorem in the literature? I judge from the responses, probably not. Aside from my data, which picks out those primes, one reason I ask is that in his Bull. London Math. Soc., 9 (1977) paper, Koblitz also relates the primes $p = 2, 3, 5$ to $j$, as follows: $p$-adic modular functions for these values of $p$ have "natural expansions" in negative powers of $j$ times a certain differential because there is "one supersingular value $\beta \equiv 0$ (mod $p$)." For larger $p$, the natural expansion Koblitz specifies involves other $\beta$ as well. Just a guess, but this seems more likely to connect to my observation than Borcherds' result, simply because it describes such a relationship.