The reactance is to help smooth out the arc and stabilize the weld characteristics. In any type of welding its very necessary.
The resistor current control will work just fine provided it gets its 12 volt power from a constant source.
Alternators failing to full on is not all that uncommon. Your battery usually takes the massive amp input and limits the voltage for a little while. The cars electronics are protected to take that level of surge just long enough for the fuses to blow. The smoking belts tends to put a fairly fast stop to the over charge situation if common sense doesn't tell you somethings horribly wrong in the first place.
Rectifiers dont limit voltage. They just work as check valves up to a peak reverse voltage where they then conduct both ways and usually burn up causing a dead short.
I am not sure how you plan on building a TIG gun being they are very application specific to TIG welding and the required shielding gas flow characteristics that go with it.
Stick and MIG (wire feed) welding systems are fairly easy to make out of an alternator for the power source but TIG is a whole different method of welding and its typical applications are far more application specific which is also why the power source has to be considerable different in its over all design as well.
I worked as a service tech for Praxair and I have the schematics for a large number of different types and brands of welding machines and plasma cutters. So if you need a referance schematic I can easily provide them for any of the common welder types.