The principal weakness shared by all of the non-canonical Gospels written in the first few centuries AD, including the Gospel of the Ebionites, is that they could not be traced to authoritative primary sources.
- Matthew is attributed to an eyewitness who was an apostle. This attribution is unanimous among early historians, is explicit already in the 2nd century (though it can be traced to the 1st century -- see my work here), and it is consistent with 100% of the manuscript titles, superscripts, and subscripts.
- Mark is attributed to the missionary companion of multiple apostles, and in the 2nd century multiple historians report that Mark wrote based on the testimony of Peter. This attribution is unanimous among early historians, and it is consistent with 100% of the manuscript titles, superscripts, and subscripts.
- Luke is attributed to a companion of the apostle Paul, who diligently acquired the information in his gospel from eyewitnesses (see Luke 1:1-4). This attribution is unanimous among early historians, and it is consistent with 100% of the manuscript titles, superscripts, and subscripts.
- John is attributed to an eyewitness who was an apostle. The eyewitness testimony is affirmed by the text itself (see John 21:24). This attribution is affirmed by a scholar one link removed from John himself (Irenaeus of Lyons). This attribution is almost unanimous among early historians, and it is consistent with 100% of the manuscript titles, superscripts, and subscripts.
The Gospel of the Ebionites has no such pedigree. The author and the authority upon which he wrote are unknown (even to the ancient historians). The text appears to date to the 2nd century when no eyewitnesses were available, whereas all 4 canonical Gospels were written in the 1st century, at a time when eyewitnesses were still around.
It was very easy for the Christian scholars to reject the authority of the Gospel of the Ebionites by pointing out these straightforward, historical facts.
Enforcing a canon, on the other hand, was not a simple task, as evidenced by the proliferation of pseudepigraphal literature in early Christianity. Prior to the 4th century there was not a centralized structure in the church capable of instituting one canon that everyone would use. By the time period specified in the OP, this centralized structure was much more developed.