'Why don't we let companies compete to offer these benefits, and let people pay for what they want?'

That's what education vouchers do, and many Republicans do support them. For national defense there is a free rider problem -- the military would protect your country regardless of whether you individually "subscribed". The general answer to your question is that the federal government largely exists to transfer money to the poor and the elderly. Big taxpayers are not paying primarily for services but for such transfers.