you can compete on price, or you can compete on quality. price competition can take various forms, from operational efficiency (better actuarial science, more efficient claims, etc.) to marketing efficiency (better targeting, simpler product, etc.). quality competition would encompass more sophisticated products/product lines, better customer service, etc.
so there is a lot of room for competition to drive down price (and increase quality) without it being a zero-sum game. the problem is that markets for financial products are irresistable to the greedy (as it's stripped down to the bare essence of pursuing money itself) making ethically dubious decisions under greedy incentive structures. the government can create a more competitive market through regulation and enforcement but chooses not to (because of its own perverse incentives), so it's not a given that a government monopoly would necessarily be any better.
with most issues like this, the default option should be to do both: have government-run options (with no unfair advantages) in competition with (for-profit and non-profit) market participants. this is a pretty rare configuration however, since it removes the obvious (unfair) leverage points for taking advantage of a market.
