had to have been vs. has to have been vs. must have been

Question

At 4:18 of this video titled "Trump's UN Speech Advocates For Both Peace And Nuclear War", Stephen Colbert said:

Now, the most awkward moment had to have been when he brought up Venezuela.

And then he showed a clip of Trump's "most awkward moment" at his UN speech.

I wonder if it has to be had to have been instead of has to have been or must have been.

I'm pretty much sure that you can easily use must have been there without any problem. But I'm not quite sure why Stephen used had to have been instead of has to have been in this context.

Answer 1

The speaker is speculating about something in the past (so basically it is a past modal here, meaning has to have been is irrelevant) and there is different degrees to speculation indeed. Had to have been is stronger than must have been and might/may/could have been is the weakest of all. Must have done means it's almost certain that something was the case while had to have done means it's very certain that something was the case.

Answer 2

I think it's because 'must have been' already has another meaning, i.e. the deductive one.

e.g: Who took the car? It must have been Paul because Jim's still here.

So to say that moment 'must have been' the most awkward would suggest that Stephen Colbert wasn't completely sure, that he was making a deduction based on less than all of the facts.

'had to have been', on the other hand, has a persuasive feel - not that the speaker is unsure but that he wants to convince his audience. This, I believe, is the same difference in feel as between:

I must lose weight - I personally feel this is necessary, internal motivation

and

I have to lose weight - an outside authority or need says so, external motivation.

(For difference between 'must' and 'have to', see Murphy English Grammar in Use, CUP 2012 or Swan, Practical English Usage)