Why does 十全十美 mean complete and beautiful?
Perhaps you were misled by the 美, thinking, naturally, it has something to do with a thing of beauty / being beautiful and so on. Not quite so as explained below.

First, "十", 10, here acts as a qualifier on 全 to indicate a 10 on a whole, (全), scale of 10.     

In other words to describe a desired state of "perfection", "completeness", "flawless", i.e. 10 upon 10.

Like 他在每件事上都追求十全十美的结果 -- He seeks perfect results in all matters.

The 美, like "beautiful" in English usage has meanings beyond beautiful in looks; like 美妙, (wonderful), 美味, (delicious) So the 美 in 十全十美 indicates an absolutely positive, "beautiful", conclusion to whatever matters undertaken.