There is a long history to the jazz violin. Chris Haigh’s website does a great job of documenting this lineage. Moreover, there are excellent contemporary jazz violinists. More interesting than thinking about this history in terms of a continuing linear evolution is considering the breaks and ruptures, and the roads not taken. On this latter note, there was a noticeable shift after the 1920s in which it seems that the (sometimes prominently featured) string quartets and (frequently prominently featured) violinists of hot jazz bands all but disappeared. By the time of big bands and the swing era in the 1930s, strings were no longer commonplace in popular jazz.
I think there were perhaps multiple factors that played into this transformation, and there are no simple reasons for the change. For example, a pat explanation is that big bands were simply too loud for violins and cellos. While at first glance this makes sense, it overlooks the important story of Eddie Lang, often credited with being the first jazz electric guitarist. The story with Lang is that he experimented in the early 20s with some of the first valve-based amplifiers made by RCA, using pickups made from hacked phonograph cartridges and telephone receivers. Already as early as 1917, the Russian scientist and cellist Lev Theremin had designed an electric cello, built by the early 20s, and presumably jazz string players experimented with methods of amplification just like guitarists.
Here are some related pictures:
The violin has a long history in American folk music.
Buskers in the early 1930s
New Orleans band from early teens featuring acoustic guitar, violin, bass.
Jazz band from early 20s, violin left and rear
Early Creole jazz band from Ken Burns’ PBS jazz series