Insulin upregulates GLUT-4 glucose transporter insertion in skeletal muscle and adipose tissue. But there are many tissues in the body that use insulin-independent glucose transporters (like GLUT-3 in the brain). Would there be any disadvantage to have the entire body use insulin-independent transporters like GLUT-3? Why involve insulin in this? Wouldn't the concentration of glucose in the blood regulate the influx of glucose via concentration gradients anyway?
The whole point of hormones is that they allow the organism to control the metabolism of tissues in an integrated manner, rather than each tissue acting independently. Insulin, glucagon and epinephrine are classic examples of this in relation to carbohydrate metabolism. In the fed state, the fasted state and in stress situations it is appropriate for different tissues to respond in different ways. The presence or absence of a hormone receptor — together with the enzyme complement of the tissue — achieves this. Allowing all tissues to have only GLUT-3 (or all GLUT-4) would mean they would all respond to changes in glucose availability in the same way — a totally inappropriate situation for the organism, with potentially disastrous consequences.
More Detailed Answer
Insulin is an indicator of the fed state. In such circumstances there will be a surplus of glucose and it is of benefit to the organism for it to be taken up by muscle and adipose tissue for storage as glycogen and fat, respectively. Conversely it is important that in starvation blood glucose is not so utilized, but spared for those tissues for which it is vital in any conditions. (Muscle can utilize other metabolites such as fatty acids.) Hence, for these tissue allowing glucose transport to be controlled only by blood glucose concentration would not produce a sharp cut-off and result in deny this resource to those vital tissues which cannot replace glucose by other energy sources.
The tissues you mention such as brain (and erythrocytes) depend absolutely on glucose, irrespective of whether the organism is in the fed or fasted state. If they had only an insulin-dependent transporter they would be starved of glucose during the fasted state, with lethal consequences. The insulin-independent transporter allows continued transport down the glucose concentration gradient, even as the blood glucose concentration falls.
Note that blood glucose concentration in the fasted state is maintained by the liver, so that this tissue has an insulin-dependent transporter, not to allow glucose uptake, but to allow the glucose produced by gluconeogenesis to be transported out into the blood.
I would hope that the logic of my answer will be sufficient convincing, but it is good practice in Biology-SE to provide references to support one’s arguments. Section 30.22 of the online edition of Berg et al. discusses the metabolic profile of different tissues and includes some discussion of glucose metabolism and glucose transporters. (The liver glucose transporter is actually different from that in nervous tissues, as is that in erythrocytes — see Wikipedia article.)