Mechanically following the binary addition algorithm yields:
1 0001 0000 → 16 1001 1000 → -24 1010 1000 → -40 (wrong!)
The binary addition algorithm does NOT work with sign-magnitude. There are algorithms that do work with sign-magnitude, and some early computers were built to use them. (And other computers were built using other not-ready-for-prime-time algorithms). It took several years of experience for computer science to decide that a better way had to be found.
There is a better way. Recall a question and its answer from several pages ago:
11111 111
0000 0001 = 110
1111 1111 = ?
0000 0000 = 010
A number added to one results in a zero. What could that number be?