Hi navi tasan :) .
I am not a teacher.
May I suggest the first example should be:-
It is best that he should be here.
The sentence implies there were other options i.e. he could be here, he would be here, he will not be here, he may be here.
That he should be here is the best option.
It is better that he is here.
Of all the other places he could have been, this is the better option.
Hope this helps.