I know very little about the history of 3D printing, except that SLA came first (in the 1980's?), and FDM development was probably held back by patents.
By 2016, very low price kit machines were available to hobbyists, in the <€300 price range, as price-reduced clones of designs which had already seen several iterations.
Was this the start of the break-out of cheap FDM machines (as opposed to the >€2000 semi-professional lab budget prototyping class), or were the earlier iterations of these kit machines also suitable/adopted by hobbyists?
I realise that early popularity would grow exponentially, but I'm thinking particularly at what point people could build a printer without needing to compile their own firmware, solder any boards, etc.
By 2016, very low price kit machines were available to hobbyists [..]
Was this the start of the break-out of cheap FDM machines
No, not by any means. The RepRap project started in 2005, and by 2008-2010 there were several open-source printer designs out there that were somewhat workable for hobbyists. These designs were still quite expensive, you needed to source all the components yourself and do a very significant amount of troubleshooting.
However, as early as mid-2009 you could buy a Makerbot Cupcake CNC for $750 as a kit (which might have involved some soldering) or $2500 fully assembled (presumably without soldering, but it's conceivable it was plug-and-play). Makerbot went on to become quite a successful company, piggybacking off the RepRap project and could be viewed as the "break-out" you ask about.
I purchased my first printer kit (no soldering or firmware involved) for $500 (plus around $150 in shipping and taxes) in February 2014; cheap hobbyist machines were commonplace well before that.