... then what are the implications when a member of the bourgeoisie advocates, say, public education?
To paraphrase Marc Faber: We're all doomed, but that doesn't mean that we can't make money in the process. Rabbi Lapin: "Let's make bricks!" Stephan Kinsella: "Say you and I both want to make a German chocolate cake."
You could argue that public education only furthers the interests of the bourgeoisie. Learnin about them great depressions and how the government is basically good so that the bourgeoisie can keep up with their scam. It also might just be liberal guilt, or an attempt to be fashionable. If the State didn't at least appear to be doing anything right, people would see that the emperor has no clothes.
People act in their perceived self-interest. Maybe a lot of members of the bourgeoisie think that public schooling "works", or that it would work if only their reforms were enacted. Or that society would fall apart if there was no "free education" in government-provided camps.
Then public education as advocated by the bourgeoisie must be bad. If it is advocated by the proletariat, then it is good.
'... then what are the implications when a member of the bourgeoisie advocates, say, public education?'
False consciouness.
'Men do not change, they unmask themselves' - Germaine de Stael
The standard Marxist response is that welfarestatist measures are taken to prevent the proleteriat from becoming revolutionary. It is essentially appeasement.
"I cannot prove, but am prepared to affirm, that if you take care of clarity in reasoning, most good causes will take care of themselves, while some bad ones are taken care of as a matter of course." -Anthony de Jasay
As someone who emerged out of the public education system to now pursue teaching in it myself, I know that textbooks do not read that the New Deal "saved" us from the Depression. They generally describe the New Deal as an effort to reverse some of the effects of the economic meltdown, namely skyrocketing unemployment. You can read between the lines all you like, and fuss about whether or not the liberal education establishment actually believes that the New Deal was the main cause of any sort of economic growth inbetween the 1929 crash and mid-WWII. However, the education establishment as an arm of "bourgeoisie" propoganda would simply state that the government cooperated with businesses, and sometimes coerced them. Essentially Wartime Keynesianism "helped" the United States to escape the Depression, but it wasn't necessarily even the main benefactor in the return to real economic growth.
I suppose these particular bourgeoisie would see the establishment of public educations as a means to achieve their ends of educating their or other children etc. However, what will happen is that it will establish an entrenched bureaucracy along with public sector unions and an ever farther spiral downwards of educational quality. All of this creates new dependents on the state, which in the end I suppose could be seen by the fabians as another step on the road to serfdom.
"Man thinks not only for the sake of thinking, but also in order to act."-Ludwig von Mises
Fabianism has some interesting aspects. On that note, perhaps one day mankind will be inclined to revisit removing the price mechanism from economics, a topic of early socialism and communism. Where's a Star Trekian replicator when you need it.
Bourgeoisie advocates public education for the masses. Sends own children to private schools. Don't mix the classes. If bourgeois children socialize with the riff-raff and street urchins from public schools they might get class indentity disorder.
Depends on how class-conscious a bourgeoisie is. Many people are apolitical in all socio-economic classes. Perhaps the most aware of their situation are the middle-class, who are pinched and squeezed and oppressed by government and corporations in kind.