This question is related to the results in

*Bergelson, Vitaly; Richter, Florian K.*, **Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions**, Duke Math. J. 171, No. 15, 3133-3200 (2022). ZBL1514.37018.

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 $N \to \infty$ for $1$-bounded sequences $a_n$ (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 (insert some arbitrary bounded coefficients in the $q$ summation, which is a summation over $m$ in the notation of that paper). 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, 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.)

I'll also remark that the quantity (1) can also be rewritten in sum-of-squares form as
$$ \sum_{d>1} \phi(d) ({\mathbb E}_{n \in B}^{\log} 1_{d|n})^2,$$
which is a more tractable form of the expression when trying to compute it; see equation (1.10) of my paper. Not unsurprisingly, similar expressions also show up in the theory of the large sieve (and the Selberg sieve).

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

*Sathe, L. G.*, **On a problem of Hardy on the distribution of integers having a given number of prime factors. II**, J. Indian Math. Soc., New Ser. 17, 83-141 (1953). ZBL0051.28008.

and

*Selberg, Atle*, **Note on a paper by L. G. Sathe**, J. Indian Math. Soc., N. Ser. 18, 83-87 (1954). ZBL0057.28502.