"Presentism" is the word— the belief that the past should be judged by the views of the present day. Kinda breaks apart when one admits that the views of the present day are constantly changing, and not always for the better.
If I am to be honest *all* monarchies should be viewed as repulsive— dictatorships with a fancy title— but for some 10,000 and some odd years of recorded history that's pretty much all there was, with a few notable exceptions— and even those sometimes were just window-dressing.
I would need to know considerably more than I do about the man and his times to label him anything, least of all a modern pejorative that is itself questionable and presumptive— "puritanical bigot" is a relative statement, highly subject to the user's on biases, assumptions and prejudices as to what is meant and to whom it applies. Indeed, watching the current news I see that bias and irrational justification happening in spades— and mostly from people asserting their own brand of self-righteous purity.
So, no, I will not label Henry V as anything. Gifted tactician and strategist, facing an incompetent bunch of swelled-headed stuffed-up show boaters who couldn't plan a Saturday night sock hop if you spotted them the playlist and dance floor. Was he "puritanical"? Were they, from a different perspective? Were they all "bigots" towards each other? Does it really matter? I don't care— I only care about the battles and who won and how.
In the end, I side with Good King Hal at Agincourt because it's easy to pick a winner who already is known to have won. And because my ancestors were largely Brits and Scots, which leaves me more muddled in my allegiances than a farmer in Normandy (potentially also a relative). And because I like Shakespeare.