If several of the players that John T. Knox mentions existed then a game or games must have been played.
This is simply untrue, and suggests a very lax standard for evidence. Historical fiction is written all the time that refers to real people and real places; it doesn't mean the particular stories are true. We know George Washington existed, but he didn't cut down any cherry tree. That story, believed by many, is a myth. Just because he was real does not mean all stories told about him must be at least partly true.
A claim should not be accepted as being likely true unless there is sufficient evidence to do so. If you do not use such a standard of evidence, you will allow yourself to believe mutually-exclusive claims to be true. For instance, Chick Murray and Henry Joseph make mutually-exclusive claims about the development of 1870s hockey in Montreal. If the fact that they both refer to people who we know to be involved means that we must accept their stories are true, we end up accepting two mutually-exclusive versions of the same events. This is not tenable, and is why we seek corroboration. If you care about whether a particular claim is true or not, you will seek corroboration.
There are ample reasons to be skeptical of Knox's story, to the degree that it can be safely rejected unless further evidence is uncovered. While some of the people mentioned can be found in censuses etc, many cannot. This is not damning in itself, since censuses around this time were not nearly as complete as later ones, but the lack of names is noteworthy.
The use of the term "rover" is anachronistic. Its first use in Montreal hockey was in the 1890s, before then there were just four forwards.
Most of the positions are named after lacrosse, but this game had only been demonstrated in Montreal three years earlier, and was certainly not codified with such positions at the time. It wasn't until the 1850s that lacrosse was organized in that manner in Montreal. So all of the position names are likely anachronistic.
The claim about inventing a rubber puck at the spur of the moment is too "just so" to be believable without corroboration. Also vulcanization was not invented until 1839, and natural rubber becomes brittle at cold temperatures and as such wouldn't be a great choice for the object of play in an ice game.
The claim that the hockey stick was born in November 1836 is also dubious. The sticks used in 1870s hockey in Montreal were imported from Halifax. Why would they do this if hockey sticks were already known in the city?
While a source document is claimed, almost none of it is actually reproduced, and apparently the entire thing was written by the younger Knox in his own hand anyway. There is no documented provenance. Nothing of it survives today. Do you really want to accept at face value the claims of a man who said that he and "lots" of his chums would have skated rings around an in-his-prime Syl Apps? Clearly Knox was an exaggerator to say the least.
And, of course, the old problem of not defining terms arises here. If Knox and his chums invented "hockey", as is claimed, what do they mean by hockey? The game described appears to have been ice hurling, in fact, and if any similar game played on ice can qualify as "hockey", we know that such games were played in England at least before 1837.
So even if everything that Knox relates in the story is accurate, the claim that it was the "first hockey" is undemonstrated. What made it hockey, and other similar games played on ice in England not hockey?
So, in summary: there are strong reasons to reject the story at face value, and even if every word of it is accurate, the claim of it being "first" seems false anyway.