This question is related to the results in

<cite authors="Bergelson, Vitaly; Richter, Florian K.">_Bergelson, Vitaly; Richter, Florian K._, [**Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions**](https://doi.org/10.1215/00127094-2022-0055), Duke Math. J. 171, No. 15, 3133-3200 (2022). [ZBL1514.37018](https://zbmath.org/?q=an:1514.37018).</cite>

Given a finite set $B$ of natural numbers, they introduce the quantity
$$ {\mathbb E}_{n \in B}^{\log} {\mathbb E}_{m \in B}^{\log} \Phi(n,m) \tag{1}$$
where $\Phi(n,m) := \mathrm{gcd}(n,m)-1$ and ${\mathbb E}_{n \in B}^{\log}$ denotes the logarithmic averaging operator
$$ {\mathbb E}_{n \in B}^{\log} f(n) := \frac{\sum_{n \in B} f(n)/n}{\sum_{n \in B} 1/n}.$$
In Proposition 2.1 of that paper, they establish the Turan-Kubilius type inequality
$$ |{\mathbb E}_{n \in [N]} a_n - {\mathbb E}_{q \in B}^{\log} {\mathbb E}_{n \in [N]: q|n} a_{n}| \leq ({\mathbb E}_{n \in B}^{\log} {\mathbb E}_{m \in B}^{\log} \Phi(n,m))^{1/2} + o(1)$$
in the limit as $N \to \infty$ (one can be more precise about the $o(1)$ term if desired).  In fact their proof shows the stronger bound
$$ {\mathbb E}_{q \in B}^{\log} |{\mathbb E}_{n \in [N]} a_n - {\mathbb E}_{n \in [N]: q|n} a_{n}| \leq ({\mathbb E}_{n \in B}^{\log} {\mathbb E}_{m \in B}^{\log} \Phi(n,m))^{1/2} + o(1);$$
this is not quite the dual of Lemma 2.4 of that paper, but can be proven by a similar method.  This gives the type of bound you seek as long as the quantity (1) is small.

In that paper it was noted that examples of sets of $B$ with small (1) include the sets of numbers with at most $k$ prime factors, for a fixed $k$.  In a [recent paper of mine][1], I worked out what the maximal size of $B$ could be while still keeping (1) small (this also answered a question of Erdős and Graham); see Proposition 2.2 of my paper.  The optimal example is essentially the same as the one worked out by you and Will, namely the numbers with at most $o(\sqrt{\log\log n})$ prime factors.  (One has to be more careful with this construction if one wants a single set $B$ which works at all scales $N$, rather than working with a single large scale $N$ and permitting $B$ to vary with $N$, but this is a technical detail.)

p.s. the asymptotic formula for counting the number of numbers up to $N$ with exactly $k$ prime factors is sometimes referred to as the Sathé-Selberg formula, see

<cite authors="Sathe, L. G.">_Sathe, L. G._, [**On a problem of Hardy on the distribution of integers having a given number of prime factors. II**](https://oeis.org/A240953), J. Indian Math. Soc., New Ser. 17, 83-141 (1953). [ZBL0051.28008](https://zbmath.org/?q=an:0051.28008).</cite>

and

<cite authors="Selberg, Atle">_Selberg, Atle_, [**Note on a paper by L. G. Sathe**](https://oeis.org/A167864), J. Indian Math. Soc., N. Ser. 18, 83-87 (1954). [ZBL0057.28502](https://zbmath.org/?q=an:0057.28502).</cite>

[1]: https://arxiv.org/abs/2407.04226