It would be nice to see a real experiment, in a wind tunnel, with both set ups. I am not the fluid dynamics expert so I am having problems with understanding this "discontinuity" aspect. It looks to me like either one would produce a discontinuity.
I like simple, and easy to make, so I stay with the flat flap concept. I make them with a slight taper to the trailing edge, mainly for looks, because I doubt if it matters much to simply round them off or leave them square. One thing I have gone to is reducing the flap chord, particularly at the root. Some planes I built in the past had close to 3 inches of flap chord at the root. I go for 20% flap chord, as a percentage of total wing chord, which is wing box chord plus flap chord. Maybe a bit less than 20% at the tip. Control forces are pretty light, and planes set up that way turn very well. No stalling due to insufficient flap chord or insufficient deflection amount.
Some time ago, I searched the web for actual aircraft flap arrangements. I should have saved that search. On one design, which was close to that European model version, I remember seeing a seal, a type of accordion seal, between the flap leading edge and the wing trailing edge box. So, the guys who designed that plane saw some benefit in sealing that gap. Trying to do something like that on a CL model would be quite an effort.