In counterfactual regret minimization, why are additions to regret weighted by reach probability?
Algorithm 1, which I refer to in the question, produces a strategy that minimizes cumulative counterfactual regret.
The motivation is to get a strategy which minimizes overall regret (as defined earlier in the paper). That's because such a strategy approximates a nash equilibrium strategy.
In other words, cumulative counterfactual regret was designed to be:

computable with a reasonable runtime; and
an upper bound on overall regret, so that minimizing cumulative counterfactual regret minimizes overall regret.

In Regret Minimization in Games with Incomplete
Information it is proven that overall regret is bounded by cumulative counterfactual regret. I'm reading through the proof currently, and will update this answer with an intuitive explanation if possible.
But for future readers who come across this, reading that proof could be a promising direction for understanding the motivation behind Algorithm 1 and the underlying formulas.