This answer addresses your question about scholarship, not the question in your title.
Yes, there is quite a bit of scholarship on the effects and effectiveness of the Montessori method. You can find a lot of it freely available; see, for example, this Google Scholar search. (Use the links in the far right column for non-paywalled full text where available. Use the "cited by" link below individual works to see more recent scholarship that references that article/book.)
A few representative recent articles from the first couple of pages of results and their citing articles:
Ansari, Arya, and Adam Winsler. 2014. "Montessori Public School Pre-K Programs and the School Readiness of Low-Income Black and Latino Children". Journal of Educational Psychology. 106 (4): 1066-1079.
Dohrmann, Kathryn Rindskopf, Tracy K. Nishida, Alan Gartner, Dorothy Kerzner Lipsky, and Kevin J. Grimm. 2007. "High School Outcomes for Students in a Public Montessori Program". Journal of Research in Childhood Education. 22 (2): 205-217. (PDF)
Kayili, Gokhan, and Ramazan Ari. 2011. "Examination of the Effects of the Montessori Method on Preschool Children's Readiness to Primary Education". Educational Sciences: Theory and Practice. 11 (4): 2104-2109. (PDF)
Lillard, Angeline S. 2012. "Preschool children's development in classic Montessori, supplemented Montessori, and conventional programs". Journal of School Psychology. 50 (3): 379-401. Downloaded from https://www.researchgate.net/profile/Angeline_Lillard/publication/225096431_Preschool_children%27s_development_in_classic_Montessori_supplemented_Montessori_and_conventional_programs/links/02e7e51c1e7bc17ba8000000.pdf (PDF)
The main "objection" I gleaned from a quick scan of the abstracts of the above is that the benefits of Montessori preschool may not be as robust for low-income African American children as for other children. I imagine there are other criticisms to be found in the wider research, but, in keeping with the principle of autoeducation, I will let you explore for yourself.
You can also have a look at the nearly-2,000 works that cite Maria Montessori's original book, the Montessori Method, here. (Use the "Search within citing articles" checkbox to easily search within these results.)
Anecdotally, my brothers and I all attended a Montessori school for preschool–3rd grade back in the 1970s–1980s; it worked well for some of us, less so for others.