No, your interpretation is not correct. In a sense it is the opposite of what you say. You might find this other question useful.
Imagine you are studying the effect of a drug on the recovery time from a disease. Imagine that the drug has no treatment effect whatsoever, but that, for some reason, women are both more likely to take the drug than men, and also more likely to naturally recover from the disease. Suppose that no other factors affect both the decision of taking the drug and time of recovery.
In this example, gender is our $W$. Note that, contrary to what you say, $W$ does affect the probability of receiving the treatment --- women are more likely to take the drug than men. By itself, this would not be a problem. But $W$ also affects our outcome. This means that $W$ is a confounder, it induces a non-causal association between treatment and outcome, so if we try to measure the treatment effect from the aggregate data it will be biased.
However, in our simplified example, we are assuming the only factor that affects both treatment selection and the outcome is $W$. So let us split men and women, and analyze each group separately. Now, given our assumptions, within each group, there are no other factors that affect both the likelihood of getting treatment and the outcome. That is, although the probability of receiving treatment does depend on covariates $W$, if you look at individuals with the same $W =w$, the probability of receiving treatment does not depend on how they potentially respond to treatment (their potential outcomes).
This is what the metaphor "as if randomly assigned given observed covariates" means. It comes from the fact that, when you actually randomize a treatment assignment, you are guaranteed the treatment assignment is independent of everything, including the potential responses to treatment---and this last bit is important for identification of the treatment effect.
In a real research context, you need to defend that $W$ actually makes your treatment "as-if random". To do that, you need to formally articulate the relationships you believe to hold between the variables you measure, and causal diagrams are a formal and intuitive way to help you decide whether your set $W$ will in fact satisfy the "as-if random" conditions, as explained in this other question. Intuitively, $W$ will render your treatment "as-if" random if it blocks the spurious associations created by common causes of treatment and outcome and if does not create any other spurious associations. This means you have no unobserved confounders besides $W$ and that the set $W$ satisfies what we call the backdoor criterion.