Is it permissible for a girl to marry the man who proposed to her, despite her family's objection, on the grounds that he is married, has children, and is older than her, provided that his wife agrees?

What you thought about your family is incorrect. Scholars have stipulated the dislike (karahah) of an older man marrying a young girl, due to his inability to fulfill her needs (iffah) and the subsequent opening of the door to evil for the girl. This is not a prohibition of marriage, but rather a mention of what is better and more excellent. Therefore, complying with the wishes of your family is an obligation that should not be disobeyed.