In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n},\mathbf{Z})$, or the "singular cohomology of a pair" $H^*((-)\times\mathbf{P}^n_k,(-)\times\mathbf{P}^{n-1}_k)$.

Denis Nardin's answer explains a suggestive analogy, and here is a followup.

It happens that several motivic cohomology groups are infinitely generated, even if $X$ is a smooth *projective* variety over a field (even for $k = \mathbf{Q}$ and $X = \text{Spec}(k)$).

How to think about this? Is there an analogy with infinite generatedness of the singular cohomology of a pair, for instance?

In other words, what should be the "moral reason" why this infinite generatedness occurs? Any analogy coming from algebraic topology would be greatly clarifying.