I agree with you mostly, but price per share does carry some information. You don't think there would be any difference if the price per share were $0.00001 as

I agree with you mostly, but price per share does carry some information. You don't think there would be any difference if the price per share were $0.00001 as opposed to $1,000,000 - even if the market cap were the same? There is a subtle difference because the price is determined by the buying and selling of the shares - and different types of people would be buying and selling the shares at these prices.