In counterfactual regret minimization, why are additions to regret weighted by reach probability? I'm reading the algorithm on page 12 of An Introduction to Counterfactual Regret Minimization. On lines 25 and 26, we accumulate new values into $r_i$ and $s_i$:

$25.\space \space r_I[a] ← r_I[a] + \pi_{-i} . (v_{\sigma I \rightarrow a}[a] - v_{\sigma}[a])$
$26. \space \space s_I[a] ← s_I[a] + \pi_{i} . \sigma^t(I, a)$

$r_I[a]$ is the accumulated regret for information set $I$ and action $a$. $s_I[a]$ is the accumulated strategy for information set $I$ and action $a$.
$\pi_{i}$ is the probability of reaching this game state for the learning player (for whom we're updating strategy and regret values in the current CFR iteration). $\pi_{-i}$ is the probability of reaching this game state for the other player.
Why do we multiply by $\pi_{-i}$ and $\pi_{i}$ to accumulate the strategy and regret on lines 25 and 26? Couldn't we just do this:

$25.\space \space r_I[a] ← r_I[a] + (v_{\sigma I \rightarrow a}[a] - v_{\sigma}[a])$
$26. \space \space s_I[a] ← s_I[a] + \sigma^t(I, a)$

It seems to me it doesn't matter exactly how much we adjust the strategy and regrets in this CFR iteration—so long as we do enough CFR iterations, won't we end up with good values for $r_I$ and $s_I$ in the end?
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.