Here we have here a detailed history of the Ethiopian Jews: Ethiopian Jewry - History of the Community from a very well credentialed source, as you can see from their home page, etc.
There is no mention there of the 1906 letter you cite, but there is a reference there that may reflect it:
By 1864, almost all leading Jewish authorities, most notably Rabbi
Azriel Hildsheimer of Eisenstadt, Germany, accepted the Beta Israel as
true Jews. In 1908 the chief rabbis of forty-five countries had
heeded Rabbi Hildsheimer's call and officially recognized the Beta
Israel as fellow Jews.
Perhaps this "Official Recognition" in 1908 was the result of your 1906 letter. The timing seems right.
Be that as it may, the history of the Ethiopian Jews along the lines you mention is well documented and verifiable. Supra (just a few snippets - there is a great deal of material there):
Once they were kings. A half million strong, they matched their faith with fervor and out-matched the Moslem and Christian tribesmen
around them to rule the mountain highlands around Lake Tana. They
called themselves Beta Israel—the house of Israel—and used the Torah
to guide their prayers and memories of the heights of Jerusalem as
they lived in their thatched huts in Ethiopia.But their neighbors
called them Falashas—the alien ones, the invaders. And even three
hundred years of rule, even the black features that matched those of
all the people around them did not make the Jews of Ethiopia secure
governors of their destiny in Africa” (“Falashas: The Forgotten Jews,”
Baltimore Jewish Times, 9 November 1979).
For centuries, the world Jewish community was not even aware of the existence of the Jewish community of Ethiopia in the northern province
of Gondar. The miracle of Operation Solomon is only now being fully
understood; an ancient Jewish community has been brought back from the
edge of government-imposed exile and starvation.....
Relevant to the source and letter you cited and their intial contact with modern Jews:
The first modern contact with the now oppressed community came in 1769, when Scottish explorer James Bruce stumbled upon them while
searching for the source of the Nile River. His estimates at the
time placed the Beta Israel population at 100,000, already greatly
decreased from an estimate from centuries before of a half-million.
Little additional contact was made with the community, but in 1935 their stability was greatly threatened as the Italian army marched
into Ethiopia. Ethiopia's ruler, Emperor Haile Selassie fled his
country and actually took refuge in Jerusalem for a short time.
Selassie returned to power in 1941, but the situation for the Beta
Israel improved little...
After taking office in 1977, Israeli Prime Minister Menachem Begin was
eager to facilitate the rescue of Ethiopia's Jews, and so Israel
entered into a period of selling arms to the Mariam government in
hopes that Ethiopia would allow Jews to leave for Israel. In 1977,
Begin asked President Mengistu to allow 200 Ethiopian Jews to leave
for Israel aboard an Israeli military jet that had emptied its
military cargo and was returning to Israel. Mariam agreed, and that
may have been the precursor to the mass exodus of Operation Moses
Over 8,000 Beta Israel came to Israel between 1977 and 1984. But these
efforts pale in comparison with the modern exodus that took place
during 1984's Operation Moses...
I personally lived in Israel throughout the 70's and 80's and met many Ethiopian Jews, who, unlike most modern Jews, are black, but today they are accepted in Israel as Jews without reservation. At the time of their initial influx in the '70's, there was some discussion about the legitimacy of their claim to being Jews that did not require conversion, but the conclusion was reached that they were authentic Jews and had rights under The Law of Return:
In 1975, Ashkenazic Chief Rabbi Shlomo Goren wrote to the Beta Israel
telling them, “You are our brothers, you are our blood and our flesh.
You are true Jews.” Later that same year the Israeli Interministerial
Commission officially recognized the Beta Israel as Jews under
Israel's Law of Return, a law designed to aid in Jewish immigration to
Israel. The Beta Israel were ready to come home.
Because much of the Beta Israel's history is passed orally from
generation to generation, we may never truly know their origins. Four
main theories exist concerning the beginnings of the Beta Israel
1) The Beta Israel may be the lost Israelite tribe of Dan.
2) They may be descendants of Menelik I, son of King Solomon and Queen Sheba.
3) They may be descendants of Ethiopian Christians and pagans who converted to Judaism centuries ago.
4) They may be descendants of Jews who fled Israel for Egypt after the destruction of the First Temple in 586 BCE and settled in Ethiopia.
Ethiopian Jews in Israel today:
More than 36,000 Ethiopian Jews now live in Israel and despite both
economic and social hardships, their community has an integral part in
Israeli society. In 1999, Avraham Yitzhak became the first Ethiopian
immigrant to earn an MD degree from an Israeli medical school. In
2011, the Israeli Ministry of Foreign Affairs appointed Rachamim
Elazar as Israel's Ambassador to Ethiopia, making him the first
Israeli of Ethiopian descent to ever serve as an ambassador for the
State of Israel. There are still many problems within the Ethiopian
community in Israel - poverty, lack of education, etc - but large
strides are being made every day.
Most interestingly, Miss Israel is currently an Ethiopian Jew: Yityish Aynaw: The first black Miss Israel
For some serious historical scholarship, you should investigate the works of The British Historian Tudor Parfitt, Books by Tudor Parfitt,who has dedicated most of his career to investigating and writing about the history of the Jews in Africa and Asia. In particular, see: The Beta Israel in Ethiopia and Israel: Studies on the Ethiopian Jews, his important collection on this subject. I don't have it at my disposal unfortunately - perhaps there you can find discussion specifically about the letter you are referring to.
Update: Upon reading Parfitt's The Lost Ark of the Covenant, written in 2008, I found that he mentions that genetic testing has shown that the Beta Israel (Falasha) lack a genetic marker that all other known Jews in the world possess, while remarkably, The Lembas, of Zimbabwe and South Africa, do possess this marker.