Not necessarily. If insurance companies were allowed to set prices based on the bank you're deposited with, then what's currently a political process could become a market-based process. Have assets that are exposed to interest rate hikes? Higher insurance rates. Have assets that hedge against changes in the interest rate? Lower deposit insurance rates. Won't let us see your asset mix? Assume the worst case scenario.

So in this universe, when you go to get insurance at SVB, you see that it costs you 1% per year, which seems high to you. You go to the insurance company's website, and see that insurance for Chase is only 0.5%, so you switch to them instead. SVB is pressured to change to a less risky strategy by the market, not b the government.

(NB I'm not a libertarian, nor am I recommending this approach; I'm just saying that in theory such a system could work.)