the only government which can fully satisfy all the exigencies of the social state is one in which the whole people participate;
that any participation, even in the smallest public function, is useful;
that the participation should everywhere be as great as the general degree of improvement of the community will allow,
and that nothing less can be ultimately desirable than the admission of all to share in the soveretion power of the State.

このように書けばわかりやすいと思います。
つまり、１文目に対する説明を行う"that"が３つあり、件の"and that"は３つ目の説明だということになります。