No, it is not necessarily the case that marginal independence implies conditional independence generally.

As a counter-example consider the following path in a causal diagram $$ Z \leftarrow U_1 \rightarrow X \leftarrow U_2 \rightarrow Y$$ Here, $Z$ and $Y(z)$ are marginally independent, but conditioning on $X$ opens this path, so $Z$ and $Y(z)$ are not independent conditional on $X$.

In the special case of marginally randomized experiments, where $Z$ is assigned randomly without reference to any covariates, then marginal independence will imply conditional independence, as you have written above. This is because there is no arrows pointing into $Z$, so conditioning on any $X$ can't open a backdoor path between $Z$ and $Y$.

No, it is not necessarily the case that marginal independence implies conditional independence generally.

As a counter-example consider the following path in a causal diagram $$ Z \leftarrow U_1 \rightarrow X \leftarrow U_2 \rightarrow Y$$ Here, $Z$ and $Y(z)$ are marginally independent, but conditioning on $X$ opens this path, so $Z$ and $Y(z)$ are not independent conditional on $X$.

In the special case of marginally randomized experiments, where $Z$ is assigned randomly without reference to any covariates, then marginal independence will imply conditional independence for any covariates $X$ that occur prior in time to $Z$, as you have written above. This is because there is no arrows pointing into $Z$, so conditioning on any $X$ that occurs prior to $Z$ can't open a backdoor path between $Z$ and $Y$. If $X$ occurs after $Z$, the independence statement is also not necessarily true (e.g., $X$ could be on a causal path between $Z$ and $Y$, causing other issues).