I claim that there are no bad sequences in C(ω₁).

Suppose to the contrary that x_α is bad. For any countable ordinal β, there is r_β in C(ω₁) such that the distance between r_β and x_α for α < β is at most 1. For any countable limit ordinal β and any positive rational number ε, there is a smaller ordinal γ < β such that all r_β(α) are within ε of r_β(β) for α in [γ,β). For fixed ε, this is a regressive function on the countable limit ordinals. Thus, by Fodor's Lemma, there is a stationary and hence unbounded set of limit ordinals on which the function has constant value, which we may call γ_ε. Since there are only countably many ε, we may find a countable ordinal γ above all γ_ε. This ordinal has the property that for all limit ordinals β above γ, we have r_β(α) = r_β(β) for all α in the interval [γ,β), since the values are within every ε of each other. That is, every r_β function is constant from the same fixed γ up to β.

Let C_β be the closed interval of values s such that the constant sequence s of length β lies within 1 of all x_η(α) for all η ≤ β and all γ ≤ α ≤ β. These are nested and not empty, since r_β(β) is in C_β. By compactness, there is a value s in all C_β. Thus, the number s is within x_η(α) for all η and all α above γ.

Thus, we may form the desired sequence r by using r_γ up to and including stage γ, augmented with the constant value s at the stages above γ up to ω₁. This sequence is continuous, and it lies within 1 of every x_η, as desired.

I guess this argument generalizes easily to other C(κ) for ordinals κ having uncountable cofinality, so that Fodor's lemma still holds.

I claim that there are no bad sequences in C(ω₁).

Suppose to the contrary that x_α is bad. For any countable ordinal β, there is r_β in C(ω₁) such that the distance between r_β and x_α for α < β is at most 1. For any countable ordinal β and any positive rational number ε, there is a smaller ordinal γ < β such that all r_β(α) are within ε of r_β(β) for α in (γ,β]. For fixed ε, this is a regressive function on the countable ordinals. Thus, by Fodor's Lemma, there is a stationary and hence unbounded set of ordinals on which the function has constant value, which we may call γ_ε. Since there are only countably many ε, we may find a countable ordinal γ above all γ_ε. This ordinal has the property that for all ordinals β above γ, we have r_β(α) = r_β(β) for all α in the interval (γ,β], since the values are within every ε of each other. That is, every r_β function is constant from the same fixed γ up to β.

Let C_β be the closed interval of values s such that the constant sequence s of length β lies within 1 of all x_η(α) for all η ≤ β and all γ < α ≤ β. These are nested and not empty, since r_β(β) is in C_β. By compactness, there is a value s in all C_β. Thus, the number s is within x_η(α) for all η and all α above γ.

Thus, we may form the desired sequence r by finding a center that works for the sequences up to stage γ, using the separability idea in your question, augmented with the constant value s at the stages above γ up to ω₁. That is, we solve the problem separately on the first γ many coordinates, and then append the constant s sequence up to ω₁. This sequence is continuous, and it lies within 1 of every x_η, as desired.