Is character a public good?
Character is certainly a private good. A lady, gentleman, or other of good character is a blessing to the home; a stalwart friend in times of need; a charitable neighbor upon whom to lean when the waves crash and the wind slashes. Persons of good character are felicitous, virtuous, splendid folk, a boon to the community.
But unless extremely unlikely events collude to test the very fabric of the nation, it is challenging to admit of circumstances in which distant people can reap the benefits of character from afar. My eudaimonia is greater for the character of my neighbors, but hardly budges at all for the character of the citizens of Cincinnati.
To a point anyway. Truly bad character that spills into wide-reaching vice, into broadly-distributed criminal enterprises saps and impurifies my precious peaceful existence, corrodes the tranquility of my domicile. At a minimum level, to an obvious (if not entirely easy to define) point, there is a public interest in sowing and tending the public garden from which character blossoms.
There is a public element to primary education. But it is likely that this element is grossly oversold. Good manners, good citizenship can be learned on the playground, and the case for the public funding (to say nothing of the public provision) of education weakens the closer the student gets to the conditions needed to achieve the self-sufficient pursuit of eudaimonia.
If you are a parent, consider carefully how you help your children achieve greatness of character.