If we have poles, shouldn't they manifest themselves on Bode plots as large valued $\hat{H}$?

Sometimes you do get large $\vert\hat{H}\vert$ associated with poles, but both poles and zeroes matter for the frequency response.

The poles manifest themselves as very large (in fact infinite) $\vert\hat{H}(\hat{s})\vert$ on the complex $\hat{s}$ plane. However, the Bode plots we've been drawing show the value of $\vert\hat{H}(\omega)\vert$ only as a function of $\omega$, the (positive) imaginary part of the complex frequency, i.e., the value as you go up the imaginary $\omega$ axis on the $\hat{s}$ plane. This regime is relevant for sinusoidal frequencies (going off this imaginary axis corresponds to transients, which matters for some applications, but we won't be treating it much).

The $\vert\hat{H}(\omega)\vert$ vs. $\omega$ Bode plot corresponds to the product of distances to zeroes from a point on the positive $\omega$ axis, divided by the product of the distances to poles from this point. So there might be a big denominator at some $\omega$, but this could be canceled by an also-large numerator.

Examples: the single-pole low-pass filter does have its largest value at $\omega=0$, which is the closest you can get to the pole (which lives on the negative real axis). The single-pole high-pass filter has one pole and one zero, and these both get larger (and their ratio approaches 1) as you slide up to very high $\omega$ on the imaginary axis.