I have feet in both camps and find it a reasonable solution! I see the immense value in creating a nice "landing page" for myself and my students. While there are a number of options available in the "Non-LMS" crowd I find myself drawn to the LMS solutions simply because tends to have a "one stop shop" for most of the tools I need. That argument, however, could be used to simply say "LMS all the way!". Instead, I tend to choose the tools that work best for me within the LMS and use external tools for important tasks that the LMS just can't handle in the same way.
Denton (2012) explains nicely about the power of cloud computing in constructivism. He works through the important aspects of a constructivist lesson such as integrating group projects, having peer assessment and critique, allowing simultaneous editing, etc, and connects it nicely with Google Docs. While there are a number of tools built directly into LMS systems for uploading documents and working in groups I still prefer the power and flexibility that Google Docs offers.
Squire and Dikkers (2012) also jump on the "Non-LMS" bandwagon and make an excellent case for integrating phones into the classroom, most of which don't connect nicely to an LMS. They cite a case study where a number of teenagers found themselves attracted to having a phone in the classroom for the instant access to information that it provides. Participants in the study downloaded, on average, 23 applications, 3 of which were directly related to learning and another 6 which were tools for the device. This tends to promote a multi-application approach, much like the non-LMS crowd utilize in picking the best tool for the job.
I start with a base LMS (at the moment, Edmodo, but Canvas is at the top of my "must try" list) and look at the tools available to me. I still have a system that beats Socrative for quizzes, so that's my go-to in that category. I use Engrade for marks until I hit upon Canvas, so we're at a tie, but the LMS solution will probably win out. Document creation and sharing is all Google Docs with no reasonable competitor present in the field. The LMS tends to be my best bet for communication, but I'm not happy with a "message board" style so Edmodo's wall style serves me well, but if I move to Canvas I'll have to find a non-LMS solution for my instant and group messages.
Thus, sitting happily on my fence, I will continue to utilize the best tools for the task at hand without overdoing the complexity for myself or my students (or at least that's the goal!)
References:
Denton, D. (2012). Enhancing instruction through constructivism, cooperative learning, and cloud computing. Techtrends, 56(4), 34-41. doi:10.1007/s11528-012-0585-1
Squire, K., & Dikkers, S. (2012). Amplifications of learning: Use of mobile media devices among youth. Convergence: The International Journal of Research into New Media Technologies, 18(4), 445-464. Retrieved from http://con.sagepub.com.ezproxy.viu.ca/content/18/4/445.abstract