Yes, I would let my kid choose his/her own toys. You really do have to let them be what they want otherwise you just can cause damage for them later in the future.
Playing with barbies doesn't automatically make a boy gay or transgender, but hey, what if they were? Even if they aren't, identity won't be found by someone else forcing behaviors on them. When I was a child, my grandfather and my parents let me play with whatever I wanted, and I fell in love with cars after my first few matchbox toys, and it's a hobby of mine to work on cars now.