The best example I can come up with is the sense of sight. Try explaining light to a blind person. They can’t experience light themselves; they have to accept the word of others that it exists. If a blind person insists that light can’t exist because they can’t experience it, then they won’t be able to discuss what we know as characteristics of light (color absorption, wave length, etc.).
No problem with 'non-Christian' J.K, it is accurate and not condescending at all to me. But the problem is the 'sense of spirit' that you are asserting exists has not been shown to exist, and doesn't seem to result in anything in the physical world that is really measurable. If the sense of spirit allowed you to receive messages from the spirit world that notify you of things that you cannot know any other way, then at least we'd have an indication of something going on. Every explanation of the 'sense of spirit' that I've heard sounds like metaphors for things we already have names for: feelings of awe, amazement, connectedness. I understand what you are saying about sight, but we have a very good understanding of how sight works and that it exists; we know light exists, we have an obvious physical mechanism, our eyes, that enables us to detect it, etc. Not so with the 'spirit'.
Also, your response has numerous applications to other things that, to me, we don't have any good evidence actually exist. People assert that they are psychic and can forecast the future or can speak with the dead, and a person not possessing these 'senses', using what I understand your argument to be, would essentially be in no position to criticize that because they don't have that 'sense', in the same way that as a non-Christian I don't have a sense of spirit and supposedly can't really assess Christianity. I don't know what you think about the validity of speakers-with-the-dead and such, but I don't buy that explanation from them, it's just an attempt to wall off their supposed powers from criticism. That's why the standard usually is, and I would bet that you apply this standard to most things in your life, what are the arguments for something being true.
You can discuss 'spiritual sense' all you'd like, I don't know which non-Christians are saying it's not allowed to be discussed. What non-Christians are likely saying is that no convincing reason or evidence has been provided that a sense of spirit actually exists. How do you know that you are experiencing 'God' with this sense? Many others feel they are experiencing Allah, and throughout history have 'sensed' all kinds of spiritual dieties and beings, an inconsistency that really doesn't apply as much across humanity as far as the five senses we do know exist; if I hear a loud boom, there aren't typically a lot of people arguing instead that they heard bells.