should i put "that" here ?

Is this "that" a necessity from a grammatical point of view?

Yes, it describes "the president."

It is the president that is responsible.

Without "that" it would read, "it is the president is responsible."

One can also use "who."

It is the president who is responsible.

Not a very good book. I suggest that word should be "who".

Yes, it is necessary for the sentence.

Another of those tired, old, silly prescriptions. Both who and that are used as relative pronouns for people. This was the case in older forms of English and also in modern English.