This is not an area I'm particularly familiar with, so please don't take my comments as authoritative.
It might. Well, it would affect it, but it's not the saturation that matters, just the resulting total flux.
This "second path" is really part of the same path as the "first path". The flux must complete a circuit, just as electric current must. The total flux in the circuit is limited by the total reluctance of the circuit, given a fixed mmf.
If someone said, "don't bother with an iron core, just let the flux return through the air", you would not expect to get so much flux in the part of the circuit where you want it. We put the iron (ferrite, or whatever) to lower the reluctance of the magnetic circuit and achieve higher flux from our PM or coil.
When it saturates, the flux can only increase as it would in free space. But if we achieve a suitable level of flux, whether the core is saturated or not at this point, may not be important.
Now, what are we comparing to what?
If we have a motor with the rotor magnets mounted in some non-ferromagnetic material, then we decide to replace part of the rotor with iron to improve the magnetic circuit, then we will get an increase in flux and torque. It does not matter whether the iron becomes saturated, because by then it will have lowered the circuit reluctance and increased the flux to a level which causes it to saturate. (*)
On the other hand if you had your motor with, say, an iron rotor which was not saturated, then decided to change the rotor material to, say, a ferrite, which was saturated by the applied field ( *) then would you gain or lose? It would depend on the level at which the material saturated. I assume we would do this replacement because the ferrite was more permeable than the iron and we hoped to increase the flux. Provided the ferrite did not saturate until the flux was at a higher level than it had been with the iron, then we are in profit (flux-wise) and the fact that it is now saturated, is irrelevant. If this were not the case, then I don't think we'd consider doing the swap. (I've put an imaginary pair of B/H curves below to show this.)
If you had a motor and you had the opportunity to replace the permanent magnets with some new stronger magnets, should you be concerned if the new magnets caused the rotor core to saturate? So long as it had gone from not saturated to saturated, you must have gained total flux. The only question would be the economic one; is the gain in flux worth the extra cost of the better magnets? Though I suppose you could then ask, can we find a better core material which will allow us to get even more flux from our new magnets by having a higher saturation level?
(* IMO this was unlikely because even a small air gap had so much reluctance that neither strong PMs nor coils could reasonably provide a strong enough field to saturate a sensibly designed iron rotor. Now you have incredibly (to me) strong PMs, high μ materials and motors of unfamiliar (to me) size and layout, so that I have less confidence in my instincts. Maybe very high μ ferrites and minute high precision air gaps could be saturated.)
Another, unlikely, possibility might be that you had a rotor which was not saturated and you tried to save weight say by reducing the cross sectional area. Then you might move from unsaturated to saturated. But the saturation would not be that important, because simply by reducing the csa you would have increased the reluctance and hence reduced the total flux.
View attachment 105811 Here the blue curve is the iron and the red is some other material with higher μ but lower saturation. If you were originally operating at H=a, then changing materials would increase B, even though the material is now approaching saturation.
If you were operating at H=c then the saturation of the new material would prevent it reaching the same B as the iron.
At H=b the change of material would make no difference. Even though the new material was pretty near saturated here, B and H are the same.
Edit: All references to "ferrites" were intended to be materials with high permeability. Looking at some data tables now, I find that ferrites are not the materials I took them for. So please ignore the word and just take it that I meant such materials.
Further I find that my belief that motors were unlikely to saturate, is wrong. I find many references to saturation in PM DC motors, brushless or otherwise.
