As I read your question, the two methods (counting text features, such as nouns; and expert judgment) differ in their intersubjectivity, or the extent to which two different people using the method in the same circumstances will get the same result. Intersubjectivity doesn't require quantification. For example, pretty much everyone who meets my cat agrees that he's black, but not because we've quantified the radiation he reflects. On the other hand, quantification through reliable measurement can improve intersubjectivity. You don't spend part of every supermarket trip negotiates how many apples you're buying because a (state-certified, reliable) scale simply measures the weight of the apples.
Now, the substance of your question is whether intersubjective methods are better than expert judgment. What does "better" mean here? I think here "better" means something like "tends to give a more accurate estimate of the billable hours required to complete projects." By definition, intersubjective methods will be more precise than non-intersubjective methods; but they are not necessarily accurate or true estimators. For example, suppose your method was to count the number of "e"s in the proposal, then multiply by 700 hours. That method would be extremely precise, but probably wildly inaccurate (unless your company does really complex engineering that takes a lot of time.)
(A counting method might have other advantages over expert judgment, even if counting is less accurate. The counting method might be easier to adjust — when it turns out that multiplying by 700 hours gives numbers that are way too high, you can modify the formula to multiply by 200 hours instead, and then test whether that works better. The counting method can be implemented in a software tool, and then anyone can use it, so junior engineers would be just as good as senior engineers at estimating the time required for a project. And so on. I'm going to assume all you care about is accuracy, though.)
So you can't say a priori (without any empirical evidence) which method is more accurate. But you can do a simple experiment to test and compare the methods. Over the next six months or so, for every project that comes through your company, keep track of the expert estimates, the estimates produced by your counting method, and the actual billable hours needed to complete the project. (Don't tweak your method to improve it while the experiment is running!) This experiment will produce intersubjective evidence that you can use to determine which method is more accurate.