Is Henry V simply designed to be a patriotic play?
You can argue this both ways, but as is usual with these sort of questions, consider it from both points of view. Henry V is principally patriotic, and attention should be drawn to the type of language used by The Chorus and Henry in his famous "Once more unto the breach…" speech, as well as his actions in moving amongst his troops and his thoughts in his soliloquies. There is considerable jingoism in the play, both in the English response to the French and in the presentation of the boorish and self-important French themselves.
On the other hand, it is worthwhile to point out the implicit condemnations of the conduct and motivations of war (e.g. the Bishops, Henry's conception of his claim to either throne, the Harfleur speech etc) and to ask: 'If it is patriotic then why, and what other functions might it have?