Yes, one attaches oneself to G-d, but that is not in conflict with the comment of Rabbi Nachman.
A better translation would be: Everyone needs to intend during their prayer, that he will bind himself to the Saints of the generation, because every Saint that is in the generation is the aspect of Moshe/Moshiach, like we find that the Saints call each other Moshe, like: Moshe, you have spoken well...
There are two aspects to the Jewish people, that of the individual (היחיד) and that of the whole (הצבור). The Moshe/Moshiach aspect mentioned by Rabbi Nachman is the צבור.
This is part of the sentiment expressed in Avot 2:4 said in the name of Hillel, "אל תפרוש מן הצבור" (Do not separate from the Tzibbur.
We learn that this aspect of Tzibbur pertains to Moshe from BeMidbar 11:21 which says:
ויאמר משה שש-מאות אלף רגלי העם אשר אנכי בקרבו כו׳
That at the innermost point of every individual from the Jewish people, they are considered as one through the inclusive, comprehensive aspect of soul, commonly called the pintele Yid which is the aspect we emphasize in the long Tachanun:
שומר גוי אחד, שומר שארית עם אחד ואל יאבד גוי אחד המיחדים שמך כו׳
This same idea is understood in connection to Moshiach from the prayer said before bringing out the Sefer Torah for the three festivals which includes the words from Isaiah 11:2:
ויתקים בנו מקרא שכתוב, ונחה עליו רוח הוי׳, רוח חכמה ובינה, רוח עצה וגבורה, רוח דעת ויראת הוי׳ כו׳
That posuk is referring specifically to Moshiach in Isaiah and so we are focusing on that aspect of Moshiach that exists within each and every individual.
And so, in the part of Likkeutei Moharan which you quoted, it is emphasizing that all of the saints in a given generation share this quality of Moshe. To put this in context, the saints of the generation are equated with the Eyes of the Congregation, the Princes of the tribes. In Aramaic they are given the title, אמרכל which means literally All say. That they speak for the whole group which they represent.
But Moshe is the one who unifies them (the Saints of the generation). He is the leader of all and is referred to in Aramaic as נשיא. That there is one individual who is the manifestation of this Moshe quality in each generation.This is what is described by Chazal as אתפשטותא דמשה שבכל דרא ודרא.
All of these ideas are emphasized in the continuation of the paragraph in Likkutei Moharan, which you left out and the paragraph which follows it.
Rebbe Nachman brings out that the gematria of Moshe (משה, 345) is also the gematria of Shilo (שילה, 345), the name used in the Prophets for Moshiach. He is explaining that this is talking about the underlying concept shared by Moshe Rabbeinu and Moshiach.
In the paragraph which follows, he goes on to say that each prayer is equated with a piece of the Mishkan. That the sum total of all the prayers from all the individuals corresponds to all the individual parts for a complete Mishkan. But the Torah explains in Shemot 40:17-34, that only Moshe, distinct among all the Jewish people, has been given the unique ability to assemble all the parts together as one single unified structure.
This uniqueness is based on the description of Moshe from the Torah (BeMidbar 12:7) which says, "בכל ביתי נאמן הוא", (In my whole house, he is faithful.)
All the individual contributors to the Mishkan must bring their specific parts to Moshe for assembly in order to reveal G-d's presence below.
And so, the intention to attach ones prayers to the Saints of the generation, and in turn to Moshe who is equated with Moshiach, is to follow that same procedure described in the written Torah.