Indeed, Bentham's principle, as he quite well intended, constituted a very powerful argument for democratic government resting on universal suffrage. [Strictly speaking, to get to that conclusion required adding the lemma that each person is the best judge of his or her own pain and pleasure, an assumption with which Bentham was comfortable but that proved a bridge too far for his godson John Stuart Mill.]
But it is certainly a bridge Mill himself passes over. Mill gives exactly this argument in the first chapter of The Subjection of Women, which is, notably, an argument for universal suffrage:
I have dwelt so much on the difficulties which at present obstruct any real knowledge by men of the true nature of women, because in this as in so many other things "opinio copiae inter maximas causas inopiae est"; and there is little chance of reasonable thinking on the matter while people flatter themselves that they perfectly understand a subject of which most men know absolutely nothing, and of which it is at present impossible that any man, or all men taken together, should have knowledge which can qualify them to lay down the law to women as to what is, or is not, their vocation. Happily, no such knowledge is necessary for any practical purpose connected with the position of women is relation to society and life. For, according to all the principles involved in modern society, the question rests with women themselves — to be decided by their own experience, and by the use of their own faculties. There are no means of finding what either one person or many can do, but by trying — and no means by which anyone else can discover for them what it is for their happiness to do or leave undone.
To a very great degree, what separates Bentham and Mill is not so much the radicalness of their views at the individual level but the differences in the way they think we should assess happiness from the impartial point of view. This is as one might expect given that the real importance in utilitarianism is placed not on who is the best judge of an individual's happiness but in how one makes the judgment of what contributes to, or is required for, the greatest happiness of the greatest number. Mill puts a considerable emphasis on the need for utilitarian judgments to go beyond what must or ought to be done (in Mill's account our assessment of good advice and of good art are also to be utilitarian) and also to take into account connoisseurship of pleasure. This does give Mill's account a less democratic 'feel' than Bentham's -- but it's not, I think, because Mill is less of a democrat at the individual level than Bentham.