The axial version is far easier to build and you can adjust the cut in speed by simply altering the clearance between magnet rotor and stator. Once built you are stuck with the radial if your cut in is wrong. The thing is a pain to mount physically with windings around the outside and you must have no solid metal in the region of flux linkage.

It makes more sense to build the equivalent of a dual rotor this way with magnets inside and outside or with magnets in a drum on the outside and a spinning steel cylinder inside.

Before the days of neo it was virtually useless to build things like this so I can't see the logic of using neos in very poor designs where it costs more to get the same output. Everyone to his own choice as always.

Which bit is expensive? if it is steel discs and you can't produce them yourself then perhaps you are right. If you include resin then that also adds to the cost.

If you are thinking in terms of what I consider the expensive bits, magnets and copper, then you will not produce a cheaper machine by using open magnetic circuits.

As for the cooling, depends on how you mount it and what protection you give the coils.

For a given cost of copper and magnet your open flux design will have a lower efficiency and to maintain the same heat then you will get less out than an equivalent dual rotor. You can always build anything efficient enough not to over heat with any design if you want to but you will need to include serious inefficiency somewhere in the line to avoid stall if you want to directly connect it to a battery. The only way to include the loss in the alternator is to adopt cooling methods that can dissipate that crazy loss. Your idea may be a bit better than a dual rotor but if you do other things to make it robust you will be little better off.

As I say everyone has their choices as to the best method, much depends on local conditions and availability of materials and facilities. This method will produce a good machine with good low wind performance but in the hands of most people will not be the cheapest or most convenient. You alone must decide.

Link is here if you want to ponder this style of construction. It may help to make an outer drum as Flux was alluding too.
The draw back with these particular ones was the high impedance output 15ohms in  star . A transformer or PWM would be required for these particular stators
http://www.otherpower.com/images/scimages/5171/seely.html

Have fun with it I did.