+1 on all that.
I would only add that, as a wrap wire, we can ignore the electrical properties (resistivity, etc.) of the wrap material and focus solely on its thermal properties. And the slight advantage in heat capacitance of Nichrome 80 as compared to SS316L is mostly offset by the fact that Nichrome 80 has a slightly higher density than SS316L, such that the Nichrome 80 wrap wire would have slightly more mass than a SS316L wrap, all other factors being equal. Taken together, these factors essentially are a wash. As an example, a 38ga Nichrome 80 wrap wire measuring 1 meter in total length has a total heat capacity of about 29.8 mJ/K (note that *total heat capacity* already factors in the mass of the wire for the given coil), whereas that same wrap wire in SS316L has a total heat capacity of 32.1 mJ/K (a difference of less than 8%). So, in theory, *as a wrap wire* Nichrome 80 has a very slightly quicker ramp up than SS316L, but whether that would be noticeable in the real world is another matter.
Note that the discussion above is only when talking about a *wrap material* because when used as a core wire, we then have to consider the differing electrical properties of the materials. And the significant difference in resistivity of Ni80 versus SS316L considerably complicates the analysis, as Carambrda noted above.