In other words, is Exodus 3:14 suggesting that the Tetragrammaton is a form of the verb "to be"? Can someone explain the linguistic path from the Tetragrammaton to the verb, "to be"?
The noun היה/הוה means "to be" (see this), if you add the prefix alef אהיה it means "i will be", if you add instead a yud יהיה it means "he will be". This is the the meaning of the Tetragrammaton according to the Rashbam exodus 3:15 (written in codes א"ת ב"ש).
It seems that most interpreters took it as form of "to be", but they disagree as to what this expression is supposed to convey.
Rashi (which almost always reflects the views of the Rabbis) interprets it meaning that i will always be with you Israel; i will take you out of egypt and even of other exiles. According to this explanation the Tetragrammaton is understood in the context of the exodus only, and the name is supposed to convey something about their condition in egypt and the exodus that will follow.
However, Rashbam, Chizkuni and Sforno understood this more broadly, they interpret it meaning i will be or exist forever. The name reveals more about the nature of god himself, and the interpretation is more philosophical, and less focused on the condition in egypt, though it conveys something about their exodus as well.
So you see that this linguistic relationship has not been lost to us, on the contrary it has been understood so by our Rabbis and later Jewish commentators for the past 2000 years.
Yes, very well-known. During prayers, upon encountering the Tetragrammaton, most Jewish prayer books note that the proper concentration is -- "was, is, and will be." -- I.e., God is above time.
YHVH This is the Tetragrammaton which may not be pronounced under any circumstances (cf. Sanhedrin 90a; Philo, De Vida Moses 3:519, 529). If this section is read out loud, this name should be read as 'Lord' (cf. Septuagint). This name denotes God's utter transcendence (Kuzari 2:2; Moreh Nevukhim 1:61). This name also denotes the creative power that constantly sustains the universe. God is telling Moses that not only is the initial purpose of creation now being fulfilled, but also the process that will insure its continual existence.