Have you downloaded and looked at the report from the direct link I gave you?
Look at Table 2
'Good Honours'
Three secondary school types had lower probability of obtaining the outcome:
• Comprehensive (coefficient = -0.38, SE = 0.08, z = -4.50, p < 0.001);
• State grammar (coefficient = -0.32, SE = 0.08, z = -3.90, p < 0.001);
• State other (coefficient = -0.50, SE = 0.09, z = -5.90, p < 0.001)
(Possibly linked to uptake with science degrees.)
First Class:
Three secondary school types had lower probability of obtaining the outcome:
• Comprehensive (coefficient = -0.29, SE = 0.05, z = -5.32, p < 0.001);
• State grammar (coefficient = -0.28, SE = 0.05, z = -5.20, p < 0.001);
• State other (coefficient = -0.28, SE = 0.06, z = -4.92, p < 0.001).
(No mention of mitigating correlations).
School type was ranked 6th of 11 out of factors correlated with good outcomes.
Less significant than year 1 outcome, course, gender, and disability.
But more significant than number of A levels, A level score, month of birth, age group or UCAS score.