Continuity of Positive Historical Judaism to Conservative Judaism

In general, the Conservative movement has a few different origin stories:

1. Zecharia Frankel in Germany founding the Positive Historical School, in reaction to liberal (“Reform”) German rabbis at a conference rejecting the use of Hebrew in liturgy in .
2. The founding of the Jewish Theological Seminary in New York by Mendes and Morais in 1886 , when it became clear that HUC could not represent all Jewry after the Trefa Banquet
3. The creation of the RA and the subsequent United Synagogue, by Solomon Schechter
4. The creation of the CJLS in the 1940s and their beginning to issue takkanot that uproot some long-held aspect of Jewish law. This marked a clear break between Modern Orthodoxy (represented by the RCA) and the Conservative movement (represented by the RA) as separate streams of Judaism (JTS graduates stopped serving in OU congregations, etc).

I’m wondering how much of a direct continuity there was throughout this whole history, from Zecharia Frankel to JTS, to the RA and the CJLS. Did the founders of JTS view themselves as within the Positive Historical School, or did subsequent rabbis at JTS look back and adopt a similar world view? Was this school viewed as being Orthodox? Did Zecharia Frankel view himself as creating a new denomination? How about Mendes and Morais? Did they view themselves as Orthodox or as members of the Positive Historical School? How about Schechter? And how about the leaders in the 1940s and 1950s that initiated the break with Orthodox?