I think Gene (Eugene P. Mosca) made a number of good points
on this problem. I one thing that I fear that may be being mixed together
is motion of a charge in a uniform B-field ( or motion of a uniform
B-field relative to a charge) and a time varying B-field. Maybe the eddy
current problems got some thinking about the different problem relative
motion of a charge and a static field.
On the original question of cutting the loop, I think that this was
answered by whoever noted that the loop that is the path of integration
does not need to be a physically loop. In the case of the cut ring, if
one wants the emf between the two side of the cut, one can use the axial
symetry to divide the results of the loop integral into the conducting
and non-conducting part. Some of the pictures in Haliday and Resnick
show single not quite closed loops.
I think that in part the emphasis on closed loops exists because
the experiments on loops predates the solid understanding of E&M and at
least in my mind the closed loop stuff is mostly a simpler ( and a
possible to understand ) example of the interrelationship of E and B.
As per Schwartz "Principles of Electrodynamics (McGraw-Hill,1972)
the trick is that ( the "_vec" is used to indicate vectors)
emf about a loop = the closed line integral of
(E_vec +v_vec/c X B_vec0 dot d l_vec)
for a stationary loop the v is zero. Then one uses that contribution to
E_vec dot d l_vec is zero so only the E_vec associated with a time varying
B-field contributes. So
emf = closed line integral ( E_vec dot d l_vec )
emf = closed line integral ( partial A_vec with respect to time)
dot d l_vec
The next trick is to convert the closed line integral to a surface
integral using Gauss's Theorem. Switch the order of differentiations and
integration. Then relate the magnetic vector potential, A_vec to the
B-field to the magnetic flux.
Up until one converts from the line integral to the surface
integral, the closed nature of the path is not important and one
could ( with great care and trepidation) deal with an open path.
For a moving loop in a constant magnetic field
emf= loop integral of (v_vec/c CROSS B_vec) dot d l_vec
which turns out to be the number of lines of flux passing sweep out
by the loop elements d l_vec which again is just d Phi/d t