Omitting a "neutral" option is in fact forcing respondents to make a choice one way or the other instead of "sitting on the fence". This can be positive if this is your intention and makes analyses easier. However it would then mean that you cannot compare your answers to standard mean scores in other studies if the scale is standardized or widely used.
I see that this is a very arguable issue after reading some discussions. What I am looking for is a sort of consensus among researchers but it seems not possible. Then, should I just apply it as I said and go ahead?
Yes - if this serves your purposes and you are not wanting to compare your data with existing data/norms. And you point this out in the methods section.