Depends on .your definition of elite. Most of the academically elite schools are private, and don't emphasize athletics to the extent that public schools do.
https://www.usnews.com/best-colleges/rankings/national-universities
That said, one can be an elite *public* school and have an elite football program. It's harder, but not impossible.
https://www.usnews.com/best-colleges/rankings/national-universities/top-public
I was in the middle of posting the same thing when I saw your post come in. Exactly correct.
let's look at the AP Final Football Top 20, and see how many are in the Top 20 Public or Private Academic Universities:
1. Clemson (61) - Public - Academic top 20 - No
2. Alabama - Public - Academic top 10 - No
3. Ohio State - Public - Academic top 20 - Yes (#17)
4. Oklahoma - Public - Academic top 10 - No
5. Notre Dame - Private - Academic top 20 - Yes (#18)
6. LSU- Public - Academic top 20 - No
T7. Florida - Public - Academic top 20 - Yes (#8)
T7. Georgia - Public - Academic top 20 - Yes (#13)
9. Texas - Public - Academic top 20 - Yes (#15)
10.
Washington State - Public - Academic top 20 - No
11. UCF - Public - Academic top 20 - No
12.
Kentucky - Public - Academic top 20 - No
13. Washington - Public - Academic top 20 - Yes (#20)
14.
Michigan - Public - Academic top 20 - Yes (#4)
15.
Syracuse - Private - Academic top 20 - No
16.
Texas A&M - Public - Academic top 20 - No
17.
Penn State - Public - Academic top 20 - Yes (#20)
18.
Fresno State - Public - Academic top 20 - No
19.
Army West Point - Public - Academic top 20 - No
20.
West Virginia - Public - Academic top 20 - No