phrases - Is "I wish I had one of those ..." correctly used in the following sentence?

Sex Education Club? I wish I had one of those when I was a student.

The bolded part actually means, I wish my university had had one of those so I could have joined . . .

But I picked I wish I had one of those . . . because it's shorter.

So my question is, would a native English speaker use that phrase in that manner (as an expression or idiom)? If not, what would be a better alternative?