Any voting system that is Condorcet compatible will comply with the majority criterion. I.e. the Condorcet criterion is strictly stronger than the majority criterion (as per your source). The Condorcet criterion also doesn't allow the center-squeeze issue. Let's take your example.
You note a system where 35% of voters prefer the left extreme (L) and 35% the right extreme (R). The remaining 30% prefer the compromise (C) as their first choice. Let's assume that those who prefer the compromise are evenly divided between left and right as their second choice. The compromise is the second choice for all those who prefer either extreme (and the other extreme is the third choice). And there are only the three options. In a Condorcet ranking, this would show as
- 35% L>C>R
- 35% R>C>L
- 15% C>L>R
- 15% C>R>L
So 35% prefer L to C (L>C>R) and 65% prefer C to L (R>C>L, C>L>R, C>R>L).
35% prefer R to C and 65% prefer C to R.
50% prefer L to R and 50% prefer R to L.
The net result is that C wins over both L and R on the first round (as the Condorcet winner). So in any Condorcet-compliant method, there is no center-squeeze.
The center-squeeze only occurs in a left/right paradigm like that when only the first choice voters are counted for the initial elimination. Examples of voting systems like that are Plurality and IRV (Instant Runoff Voting; also known as Alternative Vote and other names).