TSP is a heavy duty cleaner that cuts through grease and such, they recommend that to remove all the manufacturing oils and buffing compound from the Stainless steel.
When I did a passivation on my conical, I just went with StarSan's dilution recommendation and let that sit in the conical for 30 minutes then rinsed it. When I was ready to use the conical, I then did a normal sanitizing with StarSan for a minute or two then drained it.
If you used it at a higher strength, you do need to rinse as it's removing some metal compounds, without a rinse you'll have a thin layer of those metal compounds just clinging to the surface if you just drain and let air dry. You only let StarSan air dry when you are sanitizing it because your using a diluted strength and it's not on the metal too long.
Here is a description on passivating Stainless, I changed the text color in red and made it bold where it states that it is then rinsed with water then dried.
Stainless steel
Stainless steels are corrosion-resistant by nature, which might suggest that passivating them would be unnecessary. However, stainless steels are not completely impervious to rusting. One common mode of corrosion in corrosion-resistant steels is when small spots on the surface begin to rust because grain boundaries or embedded bits of foreign matter (such as grinding swarf) allow water molecules to oxidize some of the iron in those spots despite the alloying chromium. This is called rouging. Some grades of stainless steel are especially resistant to rouging; parts made from them may therefore forgo any passivation step, depending on engineering decisions.[7]
Passivation processes are generally controlled by industry standards, the most prevalent among them today being ASTM A 967 and AMS 2700. These industry standards will generally list several typical "types" of passivation processes that can be used, with the specific method to be decided between the customer and vendor. The "Method" refers to either the use of a nitric acid-based passivating bath (Method 1), or a citric acid based bath (Method 2.) The various 'Types' found listed under each method refer to differences in acid bath temperature and concentration. Sodium dichromate is often required as an additive to promote oxidation in certain 'types' of nitric-based acid baths.
Common among all of the different specifications and types are the following steps: Prior to passivation, the parts must be cleaned of any contaminants and generally must undergo a validating test to prove that the surface is 'clean.' The part is then placed in an acidic passivating bath that meets the temperature and chemistry requirements of the Method and Type specified between customer and vendor. (Temperatures can range from ambient to 140 degrees Fahrenheit, while minimum passivation times are generally around 20 to 30 minutes). The parts are then neutralized using a bath of aqueous sodium hydroxide and then rinsed with clean water, dried, and the passive surface is validated using exposure to humidity, elevated temperature, a rusting agent (salt spray), or some combination of the three. However, proprietary passivation processes exist for martensitic stainless steel, which is difficult to passivate, as microscopic discontinuities can form in the surface of a machined part during passivation in a typical nitric acid bath.[8] The passivation process removes exogenous iron, creates/restores a passive oxide layer that prevents further oxidation (rust), and cleans the parts of dirt, scale, or other welding-generated compounds (e.g. oxides).[9][10]
It is not uncommon for some aerospace manufacturers to have additional guidelines and regulations when passivating their product that exceed the requirements in a national standard. Often, these requirements will be flowed down using NADCAP or some other accreditation system. Various testing methods are available to determine the passivation (or passive state) of stainless steel. The most common methods for validating the passivity of a part is some combination of high humidity and heat for a period of time, intended to induce rusting. Electro-chemical testers can also be utilized to commercially verify passivation.
As someone said, I wouldn't drink the beer.