NOTHING takes the "path of least resistance". Feel free to purge that awful expression from your memory banks!
The resistance analogy works just fine; you need to draw the equivalent circuit of reluctances. In the case of a screw hole in the middle of a limb, you have core material all around it (low reluctance), with a higher reluctance path (air gap + screw in series; the screw will probably be moderately low reluctance (assuming steel), but not as good as transformer iron). So, even though the core cross section has been thinned in the area, it's still shunting most of the flux through the area, just as a shunt resistor across a higher-resistance meter, say.
Flux around a hard 90° bend, works just the same as current flow around a conductor. It bunches up along the inside corner, but it certainly doesn't prevent current flow through the outside corner. There's just a lot less there.
Now, in practical transformers, you do have the matter of saturation. This is a local increase in reluctance, in response to high flux density.
Indeed, as you approach saturation, flux tends towards taking a longer path length! This prevents flux from bunching in too tightly around inside corners, or necked-down areas (like around screw holes in the middle of a limb), and forces more to the outside of the bend, or through the air.
This would be equivalent to having a resistive material, whose resistance increases at high current density. Say you had a sheet of that goop they put in Polyfuses -- it's mostly a low resistance, but it heats up around inside corners and pushes current to the outside when it gets too high. (That would be a little too dramatic of an example though- Polyfuses are designed to exhibit negative incremental resistance. A more gentle curve would be right. The inside curve, in a magnetic core, doesn't oversaturate itself, it still carries Bsat all the while. It's just that the inside track doesn't carry much more than Bsat, and the total cross section that's sitting at Bsat grows until the whole cross section is filled in.)
It would be in this case, where you expect to see screws heating up more. Basically, the core loss on such a design, will increase anomalously by, say, Bsat/2 (for a 50% neck-down at the screw hole), and the inductivity falls; and this will continue until Bsat proper is reached, when inductivity completely tanks and the whole core becomes saturated.
Related riddle: what happens if you put steel pole pieces (Bsat = 1.5T) around a superconducting magnet (B = 4T, and assuming the coil maintains the same current flow)? Does the field increase, decrease or remain the same? If different, how much?
Tim