An auxiliary verb is by definition a grammaticalisation, since it is no longer a content word. I have spoken does not have much to do with possession, I will speak does not have much to do with desire, and I shall speak does not have much to do with obligation, and I am speaking does not have much to do with existence or equivalence.
I said “have much to do”, not “is unrelated to”: the past meanings do colour the present grammatical meanings. But you can use I will speak in contexts when you don’t particularly want to speak at all. In fact, there’s a nice passage in the Memoirs of Sylvester Syropoulos, which illustrates the corresponding change in Greek quite well: the Emperor says to the Patriarch “you want to speak in favour of Church Union”, when it’s clear the Patriarch wants no such thing. That’s Greek “want to” grammaticalising into “will”, exactly as was happening at roughly the same time with English will.
Of the auxiliaries of English, to be is more contentious, because there was not a lot of content there in the verb to being with. But there is clearly a difference in syntactic scope between I am a walrus, I am red, and I am speaking. That too is consistent with grammaticalisation.