Is it bad to not test the capability of the explainability of the student at the graduate level? Suppose I am the only instructor for a subject.

I need to conduct an exam. I am confused between the following two approaches for designing questions.

Approach I: Questions with single line answers such as boolean answers, multiple-choice questions, etc., note that the answers to these types of questions do not occupy more than one line.

Approach II: Questions that demands explanation, subjective/objective explanations, etc., 

The advantages of approach I is at least trifold

(I) Marking is objective

(II) less evaluation time 

(III) No partial marking 

But, recently, one of the elder professors of my campus, opined that approach I is a bad way to conduct exams and he told that approach II is needed because of the reason that studies of students should be research-oriented and hence explainability or presentability of the concept is very important for a student and hence approach II is preferred a lot over the approach I. 

I am really concerned about the bolded part, I mean, is it really a bad way?

Assume that the subject I'm dealing with has the flexibility to support both types of questions.
It depends on what you want students to learn.  Use open ended questions for learning critical thinking.  Either approach is good if you want students to learn facts.

Nearly always, at the graduate level, you want students learn "critical thinking" from the analyze/evaluate/create parts of the Bloom Taxonomy of learning.  Frequently this is the case at the lower levels of education as well.

Open-ended questions are necessary to demonstrate the ability to create, and can make it easier to tell if students are analyzing and evaluating.

If your goal is for students to remember a list of facts, then multiple choice questions are an efficient (but not necessarily better) way to test that.

"Understand" or "Apply" are often best addressed with open-ended questions, but could be addressed either way.