Not sure HOW one can come to the comclusion that ANY Gospel is "fake".
The Gospel of Matthew is the Gosple according to Matthew, not written by or "OF" Matthew.
The Gospel is the Gospel of Jesus, Mattew's version has been dated anywhere from 60 to 85 AD:
From wiki:
Date of gospel
Scholarship believes Matthew was composed between the years c 60 and 85 [ 71 ] [ 72 ] [ 73 ] , but a small minority claim that there is absolutely no way that it could have been written in this time given the average lifespan of most people at that time. [ 74 ] [ 75 ] Ignatius seemed to have knowledge of four Pauline epistles and the Gospel of Matthew" [ 76 ] , which gives a terminus ad quem of c. 110. The author of the Didache (c 100) probably knew it as well. [ 6 ] Many scholars see the prophecy of the siege and destruction of Jerusalem [ 77 ] as suggesting a date of composition after the year 70. [ 78 ] However, John A. T. Robinson argues that the lack of a passage indicating the fulfillment of the prophecy suggests an earlier date. [ 79 ] Furthermore the Gospel of Matthew does not mention the death of James in 62 nor the persecutions of the early Christians by Nero.
This view has been challenged by two scholars almost a century apart, The Reverend C. B. Huleatt and Carsten Peter Thiede. In December 1994, Carsten Peter Thiede redated the Magdalen papyrus, which bears a fragment from the Gospel of Matthew, to roughly the year 60 on palaeographical grounds, and thus the Gospel of Matthew could have been written by an eye-witness to Jesus. [ 80 ] [ 81 ] [ 82 ]
The Gospels are sort of "biographies", though John's is more of a eulogy because of how it was written, and as such, each author wrote according to what they were told by others ( Luke, Mark and probably Matthew) or what they saw (John).
Now, some may argue about minor details, typically because they believe or once believed these thinsg to be error proof or soemthign like that, but I don't view thme this way.
I recall the many Bruce Lee Biographies I have read and even when I spoke to people that actually knew Him ( Dan Inosanto for example) there was enough differences and "issues" that it made me understand that, with someone who was a celebrity in his time and died in the information age, if there was differences with his biogrpahy, of course there must be with Jesus ( as an example).
Heck, Bruce Lee actaully WROTE his OWN stuff in his OWN words and even that is "interpreted" amongst those that knew him, knew of him, saw his movies and though he was cool.