As the discussion on the original post and answers so far suggest, it's really critical to determine precisely the hypothesis you wish to test. Looking at the proportion of "agree" answers relative to "disagree"/"neutral" is a different question than treating the response as an ordinal variable with three levels (which may make more sense in most cases). Deciding what you really want to know is the first step.
Perhaps one flaw in the proposed analysis is that presumably your questions were answered by the same participants. If that is the case, you should take that dependence in observations into account. If the goal is to distinguish among questions, it seems that Friedman test may be well suited here. In that case you would translate the responses into an ordinal variable (e.g. -1 for disagree, 0 for neutral, 1 for agree). Then, an appropriate test to distinguish among questions can be found by looking at the list of tests in the PMCMRplus package documentation that start with "fdrAllPairs". (e.g. Conover test for Friedman, Nemenyi test for Friedman). Considering that Friedman is essentially an extension of the Sign test, I suspect this would work out well even though the response is one of only three ordered categories.
If you don't have the information on which participant answered which response for which question, probably one thing you could do is make note of this fact, and then use a test for ordinal data across independent groups, like Kruskal-Wallis, or potentially, extended Cochran-Armitage test. I'm not sure that these tests are ideal in cases where the response is just three ordinal categories. Appropriate tests among questions might then include Dunn test (1964) or Conover test for Kruskal-Wallis.
Another approach is to treat the responses as essentially binomial: Agree / Disagree, where Neutral is either ignored or grouped with one of those responses. Essentially this would be the Sign test in the paired/dependent case, or Mood's median test (or other appropriate median test) in the independent case.
For a better understanding of options, I would recommend looking up a) One-sample Sign test, b) Two-sample Sign test, c) Mood's median test, d) the other tests mentioned in this response.