I am currently writing the conclusions of my bachelor's thesis on convergence spaces and there are a couple of points I would like to make, but lack the proper references to cite in order to do so.
The first point I would like to make is that one of the starting points of General Topology was trying to axiomatize the notion of convergent sequences and hence the notion of a convergence space is much closer to the origins of topology. I recall reading somewhere that the first attempts of defining topological spaces were like that, but I can't find where.
The second point is that the category of convergence spaces is much more adequate than Top, because it has exponential objects. I know there is a whole discussion of the importance of having cartesian closed categories of spaces, but I by myself cannot argument that, because my background on category theory is really small. Therefore, I need to cite someone who knows that for a fact and can give reasons why that is.
Hopefully I am not breaking any rules by asking two different sources in the same post. I thank gratefully for any answer.
The introduction to the paper The emergence of open sets, closed sets, and limit points in analysis and topology by Gregory H. Moore could be worked into your first point.
For the second, see the paper An Initiation into Convergence Theory by Szymon Dolecki (in the collection Beyond Topology) or the book Convergence foundations of topology by Dolecki and Mynard. This thesis also makes similar points. If you want a different source, try Chapter 4: Cartesian Closed Topological Categories of Gerhard Preuss' Theory of topological structures: an approach to categorical topology.