I can give you some insights regarding the grade requirements (consider that I went through all of this in Germany, in other countries there may be a different weight on the grades).
I had an awfully bad Abitur (high school degree) and was average in my Bachelor (target school). In my Master (also target school but not like top 10, more top 20 Financial TImes rankings), I was one of the best students in my year and graduated several classes year-best. That got me: 0 strategy consulting interviews. Already when I was applying for internships half-way through my M.Sc., I did not get invited to many firms. The ones I interviewed with all made me offers but were rather inhouse consultancies and smaller boutiques.
Now, after graduating, the story repeats itself. I got invited to very few companies of which all made me offers (rather strong inhouse consultancies and external consultancies – hidden champions). The big brand name employers though did not even consider me at all. I did not get a single interview in any of the strategy consulting firms. Meanwhile, many students from my degree got invited to all of those companies (even those who were not in the top 10, 20 % of my intake). I had multiple consultants check my CV and cover letter etc. and mostly got the feedback that my past grades may make my application really difficult. I am not trying to brag about my grades later on or that I got offers from the firms I interviewed with, just trying to give you some realistic perspective on how it can be – and I am generally not some kind of genius with bad people skills or anything like that. Neither would I walk in to a McKinsey interview and just ace it all. But there is a broad spectrum of smart people (who would get hired by consultancies but are not some sort of Top 50 smartest people of their country or whatever ridiculous example people like to bring up) where, based on my experience from my applications last year, mainly those people who were consistently "good" seemed to be more likely to get the chance to prove themselves than those who were rather badly performing in the past and then outperformed "good" only recently.
For those questioning my reasoning: I am fairly certain that this is (mostly) grade-related as I got in touch with multiple consultants from some of the strategy consulting firms who were all happy to forward my application to HR where the process always ended immediately after receiving my diplomas.
But I guess this saying provides hope for the future:
Alle Wege führen nach Rom.