Although you’re mostly right, Jesus did say/teach a few things that were certainly “new”. But yes, in general, the Teachings of Jesus weren’t a lot different from Hillel.
Certainly, Jews- even very Orthodox Jews can and have beleived in a given person being “the Messiah”, without getting much more than eye-rolling from amoung the other Jews. It seems, however, that once that person had died, the continuing beleif that that person was the “Messiah” starts getting the beleiver further and further from “being a (religous) Jew”.
Here’s one Rabbi’s answer as to when Christian deviated from being Jews:
"Yet there are limits to pluralism, beyond which a group is schismatic to the point where it is no longer considered Jewish. For example, everyone considers Messianic Judaism and belief in Buddah as outside of the Jewish sphere.
Historically, any Jewish group which denied the basic principles of Jewish tradition – Torah and Mitzvah-observance – ultimately ceased to be part of the Jewish people. The Saducees and the Karites, for instance, refused to accept certain parts of the Oral Law, and soon after broke away completely as part of the Jewish People. The Hellenists, secularists during the Second Temple period, also soon became regarded as no longer “Jewish.” Eventually, these groups vanished completely.
Early Christians were the original “Jews for Jesus.” They accepted the Divine revelation of the Torah, but not it’s eternal, binding nature. Initially, these Jewish “reformers” were reliable in their Kashrut, and counted in a Minyan. But the turning point came when Paul, realizing that Jews wouldn’t accept the concept of a dead Messiah, opened up membership to non-Jews. At that point, these “Jews” experienced a total severing of Jewish identity. "
In other words- the Apostles that beleived that Jesus (before the Crucifixion) was the Messiah were still 100% Jewish.
This Rabbi disagrees:
http://www.bethhillel.com/whoisajew.htm
"
these “attackers” have wrongly characterized us as Christians whose service has “Jewish trappings” that “dishonor our own belief.” Nothing could be farther from the truth. We are Jews who worship the God of Israel as our ancestors did. In fact, our brand of Judaism dates back to the first century, when Messianic Jews were called “Nazarenes” and were a recognized branch of Judaism. This recognition occurred almost 2,000 years before Conservative, Reform, Reconstructionist or Humanistic Judaism was birthed.
We are Torah observant, Shabbat-keeping, kosher-eating Messianic Jews who believe that the prophecies contained in the Hebrew Scriptures were fulfilled by Yeshua Ha Maschiach (Salvation from the “anointed one” in Hebrew and “Jesus” in English). …By Their definition of who is a Jew, (born of a Jewish mother, not converted to another religion) King David would not be Jewish. His great grandmother was a shiksa (Gentile) named Ruth. Throughout Scripture Jewishness was passed from the father, not the mother. The founders of the state of Israel when asked, “Who is a Jew” said “Anyone who is foolish enough to want to be.” …
Within Judaism today there is much difference in belief and practice. For example reform Judaism does not believe that the Torah is the inspired Word of G-d. Instead they call it man’s search for G-d. Humanistic and Reconstructionist Judiasm isn’t sure there is a G-d. Do these positions deny them the right to be called Jews? If not, why not? Who decides? After all, Torah and the G-d of Israel are two of the core elements of Judaism. How can you deny Torah and G-d and still be a Jew? Most Reform Reconstructionist and Humanistic Jews do not keep kosher in accordance with the requirements of Torah --does this mean they are no longer Jewish? The Reform Movement just recently adopted a position that embraces and recognizes same sex relationships. This is clearly against what Torah teaches—yet there is no cry to remove their Jewisness. The Lubbovitch Sect of Hasidic Orthodox Jewry believes that their departed grand Rabbi, Menachem Schneerson, is the Messiah. For several years they have prayed by his grave and still wait for his resurrection. Mainstream Judaism still considers the Lubbovitchers to be Jewish. Why then are Messianic Jews denied the right to believe that Yeshua is the Messiah foretold in the Tanakh, and still be Jews? Why are Messianic Jews told they are “no longer Jewish”. What is the difference between Messianic Jews and Lubbovitchers? The answer, of course, is Yeshua (Jesus). Just the name of Jesus alone conjures up memories of death and persecution perpetrated “in the name of Christ” against the Jewish people over the last 1,700 years.
Unknowledgeable Jews believe that once a Jew believes in Yeshua he becomes a Christian and is no longer Jewish. It is interesting to note that in Israel, the Chief Rabbi, who is ultra-orthodox, has recently proclaimed that only Orthodox Judaism is real Judaism and that all the remaining branches are “imposters”. This pronouncement would strip ninety percent of American Jews of their Jewish heritage if the Grand Rabbi’s position is accepted. Just because someone says it doesn’t make it correct."
The Rabbi makes a good point about the Lubbovitch Sect- even though many mainstream Jews roll their eyes when this sect continues to claim that Rabbi Menachem Mendel Schneerson (1902-1994) is the Messiah- I don’t see any (including the Cheif Rabbi in Jerusalem) ruling that they are no longer Jews, as has been done with those that think Yeshua is the Messiah. The line seems to be a bit fuzzy. 
), but Hillel is considered one of the greatest Jewish Rabbis of all time.