This answer is going to have a lot of overlap with Steve Gubkin's, but maybe a bit of a different focus. I agree with him that
(1) The key issue here is understanding how vacuous cases work.
(2) Introducing a notion of "removing" doesn't help, because it introduces a new primitive notion, where set theory is deliberately built on just the primitive $\in$, and because "removing" introduces its own ambiguities.
What I want to say is that there are two goals you might have here:
(a) Teaching students what $\subseteq$ means and
(b) Teaching students how to parse mathematical language.
For goal (a), I think rephrasing the definition in as many ways as you can think of (within time limitations) is great. For some students, "removing" will resonate where the usual language doesn't, and vice versa. Pointing out the vacuous case of the empty set is great here; it is usually good teaching (and writing) to point confusing cases. Just be clear that all of the things you are saying are equivalent, and clarify why they are equivalent.
But I think you also need to think about goal (b). Students need to be able to read standard mathematical texts, and standard texts will assume that the student is comfortable with the operation of mathematical logic, more and more so as you get further into the curriculum. An introductory set theory course is often the place for students to get used to the language of logic. So I don't think you should dodge the issue of finding another phrasing, I think you should take the time to explain why $\forall_{x \in \emptyset} \ P(x)$ is always considered true.
I haven't tried this, but studies on the Wason selection task find that students understand the mathematical meaning of "if" best if it is formulated as a rule to be enforced (e.g. "If you are drinking alcohol, then you must be over 18".) I suspect the same would work with statements of the form $\forall_{x \in \emptyset} P(x)$. For example: "A website has a rule that it will only host videos if all of the humans depicted have given their consent to be filmed. Does a video which shows only cats obey this rule?"