Is it safe to reveal an arbitrary EC point multiplied by a secret key?

Assuming that by “primary order” you mean “prime order”, this is called the static Diffie-Hellman problem, and there are some known attacks against it. The main ones to consider off the top of my head would be:

• the Cheon DLP with auxiliary inputs attack: there are several variants, but for example if the prime order $p$ of your elliptic curves satisfies that $d$ is a divisor of $p-1$, then an attacker making many queries can solve the discrete log problem in time $\widetilde{O}(\sqrt{p/d} + \sqrt{d})$, which in the worst case $d = \Theta(p^{1/2})$ is $\widetilde{O}(p^{1/4})$. This could be pretty bad in principle, but there are various reasons why the attack is not extremely threatening in this setting. Firstly, due to the attack's memory requirements and the fact most values of $p$ would be fairly far from the worst case, it's likely to be hard to mount in practice in most scenarios. Secondly, if $\alpha$ is the secret key, the attacker needs to somehow obtain both $\alpha G$ and $\alpha^d G$ for $G$ the group generator, and the only way to do so seems to be $d$ sequential adaptive queries $G \to \alpha G \to \alpha^2 G \to \cdots \to \alpha^d G$. This is a lot to expect from the oracle, and since it takes time $O(d)$, this reduces the optimal $d$ to $\Theta(p^{1/3})$ and increases the resulting complexity to $\widetilde{O}(p^{1/3})$ (assuming again that $p-1$ has the required form). There aren't many settings in which the oracle would answer $2^{85}$ queries from the adversary. This is why the Cheon attack is usually more relevant in settings where group elements of the form $\alpha^j G$ are readily available as public key material, rather than obtained using sequential queries;

• the Granger extension field attack if you happen to be using curves over extension fields of degree $\geq 3$ (which is most likely not the case).

Thanks to @fgrieu for correctly pointing out that several comments I made about this were incorrect, and also pointing to relevant discussions on an IETF mailing list.