Reuvain actually offered to be responsible for Binyamin even at the cost of sacrificing his own children, but Ya'akov rejected his offer. The Tiferes Shlomo explains Yehudah's words "anochi a'arvenu" had profound meaning to Ya'akov. The gemara (Baba Basra 173) learns from the words of Yehudah that there is a halacha of arvus - if person X wants to make a loan, I can become an areiv by guaranting my assets as collateral in case the borrower defaults. On a metaphysical level, Jewish survival is based on our each bearing responsibility for each other, in effect offering ourselves as collateral to see not only that we each grow in mitzvos, but that those around us are not left behind. Outside the financial realm, halacha has another principle of arvus which allows me to be motzi someone else in a mitzvah even if I have already done it myself - my mitzvah is incomplete until the next person also has been yotzeh. Chazal tell us Ya'akov instituted tefilas "arvis". Tiferes Shlomo explains that this word, commonly translated as night, also comes from the root of "areiv", as our co-responsibility is most needed in the dark night of galus. When Ya'akov heard Yehudah use this formula, he knew Yehudah would not lose Binyamin, My wife added a beautiful chap based on the relationship between arvis and arvus. She suggested that the common practice of women not dabening arvis at night may be related to the machlokes haposkim (R' Akiva Eiger, Dagul M'Revava) whether the principle of arvus for mitzvos extends to women as well, e.g. can a women who was already yotzeh a mitzvah perform that mitzvah again to be motzi a women who was not yotzeh already.
I would suggest that Yehudah's promise of "anochi e'ervenu" (he will be an areiv) rang true because he already had shown his fidelity to truth at all costs by confessing to be the owner of the "eiravon" which he left with Tamar.