Whenever input A is set to 1 you want output A to light as shown in the diagram below. There's nothing wrong with output A lighting because the diagram shows that's exactly what you want.
The reason output A is a don't care for 001 is because you stated you will never have 001 as an input. In your testing, you see that output A is lit when input A is set to 1 because that's exactly the functionality you're after.
The purpose of the don't care is to simplify the amount of logic you need. Which it has done here. You don't have to special case output A to worry about whether B and C are set when A is set. For input A, output A only cares about what input A is.
If you cared about 001, then your output logic for output A would be:
A = B + CA
A = B + A
You would need an extra AND gate for C*A. Having that don't care reduces the amount of gates you need to get your output properly.
Lastly, it would be a bit less confusing if you used a different naming convention between input and output. Something like input = XYZ and output = ABCDEFG or input = ABC and output = abcdefg.