Your logic applies in exactly the same way to the good old one-sided tests (i.e. with $x=0$) that may be more familiar to the readers. For concreteness, imagine we are testing the null $H_0:\mu\le0$ against the alternative that $\mu$ is positive. Then if true $\mu$ is negative, increasing sample size will not yield a significant result, i.e., to use your words, it is not true that "if we got more evidence, the same effect size would become significant".
If we test $H_0:\mu\le 0$, we can have three possible outcomes:
First, $(1-\alpha)\cdot100\%$ confidence interval can be entirely above zero; then we reject the null and accept the alternative (that $\mu$ is positive).
Second, confidence interval can be entirely below zero. In this case we do not reject the null. However, in this case I think it is fine to say that we "accept the null", because we could consider $H_1$ as another null and reject that one.
Third, confidence interval can contain zero. Then we cannot reject $H_0$ and we cannot reject $H_1$ either, so there is nothing to accept.
So I would say that in one-sided situations one can accept the null, yes. But we cannot accept it simply because we failed to reject it; there are three possibilities, not two.
(Exactly the same applies to tests of equivalence aka "two one sided tests" (TOST), tests of non-inferiority, etc. One can reject the null, accept the null, or obtain an inconclusive result.)
In contrast, when $H_0$ is a point null such as $H_0:\mu=0$, we can never accept it, because $H_1:\mu\ne 0$ does not constitute a valid null hypothesis.
(Unless $\mu$ can have only discrete values, e.g. must be integer; then it seems that we could accept $H_0:\mu=0$ because $H_1:\mu\in\mathbb Z,\mu\ne 0$ now does constitute a valid null hypothesis. This is a bit of special case though.)
This issue was discussed some time ago in the comments under @gung's answer here: Why do statisticians say a non-significant result means "you can't reject the null" as opposed to accepting the null hypothesis?
See also an interesting (and under-voted) thread Does failure to reject the null in Neyman-Pearson approach mean that one should "accept" it?, where @Scortchi explains that in the Neyman-Pearson framework some authors have no problem talking about "accepting the null". That is also what @Alexis means in the last paragraph of her answer here.