Because she's MAGIC. And (Alex explanation of awesome spirit powers being blocked by negative emotions here).
Seriously, Denzel being actually sent by Aerith or not is actually sorta irrelevant, as is Cloud and Tifa thinking she brought him to them, outside of how they view Aerith. The only reason it's there is so that it can be 'Not to either of us, but to both of us' with the smile in return. Denzel's role is to be a child that Cloud and Tifa can do right by, to make up for the sins of their pasts and the lives lost directly or indirectly as a result of their deeds as AVALANCHE, and to cement them further as a cohesive family unit, by being a child who is exclusively theirs to look after, which cannot be said of Marlene, who they are looking after for Barret. He also acts as something of a MacGuffin to the plot, being the reason Cloud went walkabout initially- so he could find a cure for Denzel.
But yes, Denzel is there to be Cloud and Tifa's kid, and they certainly crammed a lot of similarities between C and T and his real parents into CoD and ACC, right down to holding his hands alike.
They showed us a photo of his old family just before he calls Tifa and Cloud finds him, to show us he's getting a proper new family.
In short, Denzel exists to hammer in the concept of the Strife family unit, that Cloud has responsibilities, people he loves and who love him, and who he needs to be there for.
On the subject of canon, recall that there is a distinction between canon and within continuity, and multiple different types of 'canon heirarchy' as well, such as the SW structure. It's entirely possible to negate all but one part of a story and keep it in while sacking the rest, as seems to be done with LO and the C/T convo therein.