I first heard this aria sung, as above, by Luciano Pavarotti - and this is, for me, the greatest performance overall. But the role was clearly Not written for a tenor of Pavarotti's type - with such a full, ringing and loud voice. (This is indeed known from multiple other sources of evidence.)
Therefore, despite that Pavarotti sings every other note with his usual, perfectly even and glorious tone, he cannot reach the high F without switching to a falsetto (head-voice) voice production which marks a qualitative break in tonal quality. The run-in to this note is from 4:25. He does the high F beautifully - yet it sounds like someone else is singing that particular note.
Pavarotti had probably the best high C of any full-voiced Italianate tenor - and what is more he possessed equally fine high C-sharp and even D - as may be heard in this aria; yet there are extremely few tenors of his type that can manage these notes using the same mode of vocal production as the high C.
But the high F is three semitone above D - and that is a long way when singers are being judged by exact standards - and way beyond what any big-voiced tenor could produce in his normal vocal tone.
You can hear this from a compilation of recordings of this notorious note - which is, I think, by far the highest note in the mainstream operatic tenor repertoire - since it is very seldom any tenor is required to go above high C at most - and that only once or twice per opera.
Ignore the drivel in the comments! - What you can hear is that any tenor who has the kind of loud, ringing tone of a Pavarotti - someone like Gedda - must change to a qualitatively different tone for the high F.
This is because all tenors have a 'break' in the voice, above which the tone must become falsetto; this break can be raised by training - but it is because of this that full-voiced tenors will sometimes 'crack' on high notes. This is like a yodel, and for the same reason - the voice suddenly, but uncontrolled because accidentally, flips into falsetto.
(Yodeling is a controlled flipping back and forth between falsetto and the ordinary voice.)
So, is the high F 'impossible'?
There are tenors who, instead of having an abrupt break, gradually introduce more and more falsetto - their voice gradually and evenly changes from normal to falsetto as the notes get higher. (I have a friend with a naturally deep and sweet-toned voice, who does this spontaneously - and who has been able successfully to sing bass, baritone and tenor roles in Gilbert and Sullivan!) For such tenors, this high F may have more, or less, falsetto according to how high the break occurs. Some tenors have a very high break, and therefore the high F has less falsetto.
However, these tenors invariably have a 'lighter', quieter and less ringing kind of tenor voice than the likes of Pavarotti and Gedda (or other greats of Italian Opera such as Caruso or Gigli) - they are, in essence, a different kind of voice (leggiero or tenore de grazia are some of the terms) - and such tenors are nowadays most often seen in Rossini, because only lighter voices can manage rapid coloratura (decorative passages of many quick notes). Luigi Alva was of this type.
It seems certain that Bellini was writing for a just this type of leggiero, ligh-voliced singer, who used falsetto-flavoured production for the high notes. However, the tenor plays a man in nearly all operas, what is more the hero; and so he needs to sound masculine in vocal quality.
This need for a heroic quality can be a problem for most high-voiced tenors, including who can best manage the high F with the minimum of falsetto. For perfection, the tenor should be able to integrate the high F with the whole of the rest of his voice - so that there is a completely-seamless transition in vocal quality.
An example of musically-desirable seamless integration is Bruce Brewer (from 4:20):
However; Brewer sounds too-much like a female contralto - and not at all
heroic!
So there is a problem. In a purist and strictly musical sense the tenor role in The Puritans must be a trade-off between heroic qualities in the voice, including the capacity to produce thrilling - rather than merely sweet - top notes; and the ability to sing the highest notes without a break in the voice. And, realistically, this can only be done by sacrificing the high F! - i.e. by singing it in (more or less) the different falsetto tone, as done by Pavarotti and Gedda.
Or else there ill be a 'compromise' between heroic qualities and the vocal production. To my ear, an example of a good compromise is William Matteuzzi (who is also included in the compilation video above). Start at 3:20 - but the high F comes shortly after 4:25.
Matteuzzi - who I only recently discovered - was a very remarkable singer in many ways, although he never made it 'to the top' for whatever reason (e.g. he seems to be a poor actor and to lack charisma, in the YouTube examples I have found).
His voice is light but it has that thrilling, masculine 'ring' (sometimes termed squillo) which is so valued by opera-goers - especially in the hero tenor roles. And this includes his highest notes which, although they are flavoured with falsetto production, have a genuinely heroic and exciting quality.
His high F is very
nearly, albeit not
quite, tonally-integrated with the rest of his range - but as soon as he steps down one, then two, notes down from the F (
E-flat, D-flat), his full 'normal' thrilling tone returns - and the effect of this passage of three notes is excellent.
In the end, I would be forced to say that Bellini made a mistake by including the high F in this role - because it is actually impossible to sing in a wholly-satisfactory way. But it has certainly led to a lot of enjoyable fun and games for tenors trying to cope with it - one way or another.