The fact that the right shift for a negative integer gives the floor function of the result just makes the correction easier than if you had used division with truncation towards zero.
The shifted out bit is always positive, regardless whether the shift had been applied to negative or positive numbers.
Except for following a tradition generated by a random initial choice, programming would have been in many cases easier if the convention for the division of signed numbers would have been to always generate positive remainders, instead of generating remainders with the same sign as the quotient.
I do not see where this would be of any use.
On the other hand, if you want a quotient that has some meaningful relationship with the ratio between the dividend and the divisor, there are other more sensible definitions of the integer division than the one used in modern programming languages.
You can have either a result that is a floating point number even for integer dividend and divisor, like in Algol, or you can define the division to yield the quotient rounded to even (i.e. with a remainder that does not exceed half of the divisor).
In both cases 10/3 and -10/-3 would yield the same result and I can imagine cases when that would be useful.
For the current definition of the integer division, I do not care whether 10/3 and -10/-3 yield the same result. It does not simplify any algorithm that I am aware of, while having a remainder of a known sign simplifies some problems by eliminating some tests for sign.
