So after the victory Alexander and Francis' eagerness would continue. I'm guessing Kutuzov would get retired for continuing his advice of caution and saying Austerlitz was luck. Plus Alexander probably would say that Kutuzov had nothing to do with the victory considering Kutuzov pretended to sleep through the war council.
alexander never liked kutuzov
so yes most likely a victory at auterlitz would be worst than better for the coalition because even if alexander would be probably become an arrogant general he would probably be like this
Alexander after presenting a shitty battle plan,and ignoring kutuzov's sugestions.
"general kutuzov is too defencive if i had followed what he whanted in austerlitz we would had never defeated napoleon"
Affter losing the battle and leaving half the russian army dead in the field
"its kutuzov's fault that we lost my plan was perfect"
in the end a defeat helped the coalition in the long run i can't image alexander allowing napoleon to come to moscow without first killing is own army in an attempet to stop the french