These are all valid and good points. But it isn't the end of the story...
Your real name will be linked to the entity for as long as you are a participant in the entity. If you sell your interest to another party (a trust perhaps), you are no longer associated.
A note about that: while you are associated, it is public record. Anyone who inquires - will be informed.
Your SS# is required when forming the corp and when obtaining the EIN.
When asked to provide SS# for things such as credit applications, the SS# can be the number of ANY principle party. Those applications are considered optional and elective. They do not disclose your info to outside parties. And the number is for file reference. Way back when, credit bureaus decided that SS#'s being unique, would be a good number to base a credit file upon. That is the only reason it began that way. Any 9 digit unique number will do the job.
A corporation whose principal parties is composed of non-persons won't have an SS# to contribute. Thus, if you are the only principal party, you offer your own SS# to the bank.
But in the case of many corporations (let's say the Heinz food group), who's SS# is on the corporate bank account?
If one of the principal parties happens to be an LLC named Roger Hammersmith, and you provide Roger's EIN...
You are only obligated to provide an SS# for personal bank accounts and any interest bearing investment. For corporate checking, there are alternatives to giving a principle SS#.
There are exceptions (usually well hidden) which provide for alternatives. I will not elaborate any further on the details for reasons OBVIOUS to many of us in here.
I'm trying to convey that there are other answers, beneath the obvious. And if you only read the bold print, you don't learn the tips and secrets in the fine print of the back pages.
Your efforts and the difficulties along the way to achieving your level of 'comfort' depend on how much anonymity you need.
 | 
