First of all, I am not good at English but I have to mention this. İt's annoying me. Please excuse my usage of language.
''It is nice to see you.''
What is the subject of this sentence? What is the thing that is nice? Obviously, it is ''to see''. To see you is nice. But according to the sentence above ''it'' is subject of this sentence. But according to the reasoning, to see is the subject, too. In the final analysis, we can say that we have two words that have the same meaning. As long as two things have the same meaning we can use the one instead of the other one. Therefore, if ''it'' is the same thing with ''to see'' then we can say that to see is nice to see you.
Or let me ask you this: Why do you use such a structure? And what do you think about it?
Best Answer
I didn't look up all the previous questions on the topic but I think I remember reading the term indefinite pronoun. You might as well say "all is nice seeing you". all is necessarily underspecified, because most of it unknown.
However, I want to note that this escapes logical analysis, because it's intuitive, colloquial speech, learned naturally with childish logic. Not to underestimate children and the aptitude of their parents, but a formalized, prescriptive, compulsory education as we know it is kind of a new thing. The question should really be where the expression is from.
It exists similarly in Germanic languages, in Russian and French, at least. I can only speak to German and French, for the others I only looked at Swedish "it rains" (det regnar) and Russian "it's cold" (tam xlad, and that's not standard). Not that Ru. tam and Proto-Germanic *þat (Sw. det, En. that), are from Proto-Indo-European *tód, nominative and accusative singular neuter of *só ("that").
The French really go through with all the ce in qu'est-ce que c'est "what's that there". There were malapropism or mishearing, rebracketing (I forgot [1]) involved in the development of French question phrasing, too (so much for naive language). ce is actually said to come from Old cil, From Vulgar Latin *ecce-illum, related to Classical Latin ecce (an interjection look! see! here!) and eccum, etc. I think our it is pretty much comparable in this use, a demonstrative pronoun. "that is nice to see you" would for some reasons remind of conjunctive "that" introducing subclauses. I mean that is reminiscent of and, another conjunction in "and, how are you?". Also compare "look at you, look who's there". Incidentally, it and here derive from the same root.
Not to mention that *so (see above) reminds not just a bit of English so (PIE *swe~se) which works as a largely meaningless emphatic sentence introduction as well (cp. It is such a nice day, and Ger. So ein schoener Tag ist es).
On a very basic level, your interpretation it=see is feasible, if you use the nominalized form: seeing is nice, seeing you; that's a minimal context, opposed to an all encompassing one. For it's simplicity, it might have a particular value. Asking what is "it" might reveal a lot of subtleties.
You have to wonder where the proscription of naming oneself first (in a list of people, mostly) comes from. Obviously that's a form of courtesy.
[1] probably heard it, though I only remember Canadian French mentioned, in Lexicon Valley, in a recent episode on the letter T