The question is "What are the reasons Artificial Languages should not be under the scope of Linguistics" (it's not a question about writing systems). As a question about personal opinion it is not suited for SE because it's not a question about fact, but that can be remedied: what reasons can be given that ALs are not within the scope of Linguistics? That is distinct (even if related to) a question about what is on-topic for LSE (such a question belongs on Meta).
The answer derives from a view of what the subject matter of linguistics is. If you define linguistics maximally broadly as being about any kind of communication and systematic transmission of information, then AL's would certainly be in the scope of Linguistics. Linguistics would encompass many disciplines, such as genetics (the grammar of DNA), bee dancing, bird song, maybe even how viruses work. Historically, the science called "linguistics" (Sprachwissenschaft, nyelvészet and so on) has been associated with a narrower set of questions, about human languages (yes, I know that we have not yet defined "language").
The dominant trend in linguistics has been to study those properties of language that occur universally and naturally, without conscious human intervention, and linguistics has been seen as a scientific discipline which asks "what is the nature of that {cognitive faculty/form of behavior}?". This narrows the scope of linguistics, so that the study of articulatory behavior of world-class opera singers is outside the scope of linguistics (Being Martti Talvela is not universally available and naturally available, it requires massive effort and inherent talent). Or, the study of theories of judicial interpretation w.r.t. what the words of a law are said to "mean" is not a part of linguistics. That does not mean that linguistic tools cannot be used in such studies, it means that the product of such research does not address the presumed central question of the science of linguistics. On those grounds, AL's would not qualify as within the scope of linguistics. You can use the tools of linguistics to construct an AL, but ALs don't inform you about the nature of languages (as defined). Of course the discussion changes once an AL is spoken so prevalently that children actually acquire it as their language.
You can disagree with the initial premise and offer an alternative statement of what you think the science of linguistics is about, and that statement could well include AL's, poetry, pheromone transmission and so on. ALs and writing systems are on the margins (both being artificial). The only argument for a particular definition of a pattern of human activity ("a study") is historical precedent. You could say, if you want, that "Linguistics" is a very broad discipline that includes DNA studies, and define the study of the universal naturally occurring human code of proposition-articulation as "nyelvészet". But there is no objective way to establish the "intrinsically correct" name for a specific field of study.