The expression is not in base-10. You then have to add everything up and put it in base-10. You find it easy to add things this way because you are used to base-10 operations. If you were used to adding numbers in other bases, you might just as easily dismiss the following addition: . This process is the exact same process you used for converting the binary number to base-10, but it might not look right because it is not in base-10. In base-10, the process we are taught for addition is to add up a column of digits, record the units place of the sum and "carry" the rest of the digits one column to the left. This is the same as taking the remainder when the sum is divided by 10.
So, you can certainly convert directly from one base to another, so long as you are comfortable with the arithmetic operations of that base.
Edit -- Here is an example of going from base-2 to base-7 without stopping at base-10: