It's to make traceability and error-recovery possible, possibly even defeat hacking and abuse. It's absolutely indispensable for community deletions and even makes sense for self-deletions. Consider this scenario: A user who wrote many valuable answers throws a tantrum and decides to go on a delete spree as retaliation. With a “soft delete”, it can easily be fixed.
The basic idea is that everybody can edit the site and gaining special rights is relatively easy but nothing is irreversible, Wikipedia-style.
Moderators (and high-rep users are effectively moderators; in fact, the relevant privilege is literally called “access to moderator tools”) and staff can still check and reverse any decision, fix mistakes, etc. The logic and value of this is quite clear! Preserving our ego is not considered a good reason to complicate that and the original contributor has no special rights in this regard (your text is an under an open-source license anyway). Edits do not erase previous mistakes either, a full history is available (and I believe comments - while considered less important - are actually archived in the database too).
Also, that way, the semantics of the “deletion” are simple and clear and stays the same in all scenarios. Since an answer can be deleted based on a mere 5 votes from random users with sufficient rep, it's difficult to see how it could be otherwise for regular deletions. You need a whole other set of tools on top of that for the “hard delete” because you can't do it by default. So if answers deleted by their authors were somehow hidden to more people, you would need another level of privileges.
I can understand your embarrassment but that's not really a compelling reason at all to add something like that at all, that would make no sense. As in Wikipedia's case, the real need for a “hard delete” only arises for legally problematic things like copyright infringement, libel, racists comments, etc.
Also, don't worry too much: Deleted answer don't show up in search results, etc. so stumbling upon them is unlikely and the number of users with the right privileges is actually quite limited. In all likelihood, nobody even saw/read it (beside perhaps the people who noticed it before you deleted it in the first place). And it's no big deal, we all have an answer somewhere we wish we had gotten right the first time.