Many of us including myself have successfully controlled bg levels by carb restriction. I want to discuss if low carbs, lesser insulin can cause body to be less nourished given that insulin is an important anabolic hormone that drive many nutrients into our cells.
If the question is Do we need insulin? then the answer is yes (it is analogous to the spark for petrol in a combustion engine). Do you need to eat starchy carbs to generate insulin. No. The lower carb you go the more gluconeogensis goes on (conversion of protein into glucose) and non starchy veg still require a little insulin to digest them.
As has been pointed out we are evolved to work on minimal levels of insulin and longevity studies seem to show that the less you have, the better but there is a physiological limit to how low this can go unless you are a type 1.. Those modern and traditional peoples who do well on a high starch diet e.g. Katarvans are acutely insulin sensitive probably because they eat non processed food and do not have huge blood sugar spikes that the majority of people eating a modern diet will get unless they are a) genetically blessed b) professional athletes.
Sadly there seems to be no easy way to measure basal insulin to check that you are insulin sensitive before we run into disease processes so we have to go for proxy measures e.g. thickening waist line, hypertension , fatty liver and eventually type 2 diabetes.
Re brain and glucose - the body preferentially uses glucose for the brain but can adapt to a mixed fuel of both ketones and glucose. Hence even pure carnivores are converting some of those steaks into glucose for use by the brain.
As a long standing type 1(with no endogenous insulin I assume) I am always amazed at how much basal insulin is needed ! On a low carb diet it is always so much more than my bolus doses (a ration of 3:1 unless I've been eating too many Easter eggs).