This should be a comment, but it's too long -  here's a $\mathsf{ZFC}+\mathsf{CH}$ example:

Let $\mathfrak{F}$ be the set of full-measure subsets of $\mathbb{R}$ *(we could also take the set of non-meager subsets, or so on)*, and let $\operatorname{Space}(\mathfrak{F})$ be the topological space whose points are the full-measure subsets of $\mathbb{R}$ and whose topology comes from the"canonical generator" $$\rho:r\mapsto\{X\in\mathfrak{F}: r\in X\}.$$ For this space **instances of covering correspond to non-null sets**: we have $\rho(g)\subseteq\bigcup_{f\in U}\rho(f)$ iff $g\in U$ or $U$ is non-null. 

A standard transfinite recursion argument shows in $\mathsf{ZFC}+\mathsf{CH}$ that for every size-continuum set of non-null sets $(N_i)_{i\in\mathbb{R}}$ there is a non-null set $B$ such that $B\not\supseteq N_i$ for any $i$. This implies a weak form of non-quickness for $\operatorname{Space}(\mathfrak{F})$, namely that the canonical generator $\rho$ does not witness quickness, and it's not hard to extend this to arbitrary generators. So $\operatorname{Space}(\mathfrak{F})$ is not quick.

(Annoyingly I don't see that if a space is quick then all generators should witness that; luckily, that isn't an issue here.)

***

Of course this example does **not** work under determinacy, since every non-null set has a non-null $F_\sigma$ subset and there are only continuum-many of those. But I suspect a more complicated ideal of sets than the null (or meager, or etc.) ideal will do the job. Note that given an $I$-indexed subbase for a $T_0$ topological space $\mathcal{X}$ there is a canonical homeomorphic copy of $\mathcal{X}$ whose points are subsets of $I$, so this isn't really a significant shift.

At the same time, the above raises a separate question:

> What can we say, in $\mathsf{ZFC}$, about the minimal cardinality of a set of non-null sets $\mathfrak{A}$ such that every non-null set contains an element of $\mathfrak{A}$? *(Or non-meager, or so on.)*

I don't really have anything to say on this point, but I'd be interested in information about it (and may ask a separate question about it later on).