Would Maine South have had a 77-game Central Suburban South winning streak if Loyola Academy and private schools did not exist?
In my opinion, no. Imagine the strength of New Trier (though New Trier West most likely would still be open), Glenbrook South, and Evanston.
Three are four private schools (Loyola, Regina, Christian Heritage Academy and North Shore Country Day) within New Trier's district. CHA and NSCD are tiny. Regina is around 350 girls, and Loyola has roughly 2000 students. Not sure about CHA and NSCD demographics, but Loyola and Regina draw a substantial majority of their students from outside of the NT district. I highly doubt that NTW would still be open as a separate four year high school if there were no private schools.
Of the schools you mentioned, GBS would likely benefit the most from a football perspective if private schools didn't exist. Year in and year out, Loyola's biggest feeder school is OLPH in Glenview, from which it always gets a ton of good players. Most of those players would attend GBS (and a handful to New Trier and Maine East) if private schools did not exist.
Sure, ETHS, NT and GBS would probably have stronger programs, but Loyola and Notre Dame enroll a fair number of kids from PR and the MS district, so most of those kids would have attended MS in the absence of private schools. Would the net effect have been enough to keep MS from having a 77 game winning streak? Maybe, but I think they still would have been the team to beat in 8A during their long run.