Yes, it should be very cheap and any sane person will also agree that this is something incredibly useful. Just if you give something to the state it becomes hard to get rid of it again, even when it becomes prohibitively expensive in the future.
When you regulate everything via competing insurances this will be better. You are free to choose your insurance and they are free to specify which certificates they will accept.
Yeah, that's why regulations must be restricted and limited in scope to things that are really necessary and not excessively burdensome to those that aren't likely to harm others.
The level of regulation for cars seems pretty reasonable. I imagine most people do not want habitual drunk drivers driving or irresponsible gun owners that regularly threaten to shoot others (outside of self-defense) and habitually leave their loaded guns laying around little kids to have a gun.... or a terrorist (whether they're Political or Religious extremists like Nazis, Al Qaeda, etc) having a gun. Focusing on the insurance, incentives, etc on a large part of the effort seems to be a good idea though.
Then we could implement things like 'sunsetting old laws' to limit the ridiculous persistence of old laws that make the law so bloated for no good reason: https://www.yang2020.com/policies/automatically-sunsetting-old-laws/
It's extremely insane that victimless behaviors like possessing marijuana are illegal right now. That's even less harmful than alcohol. People are forcefully jailed and taken out of the workforce, for a victimless crime... ruining their entire life and harming the nation altogether. I've never even smoked them, but it's ridiculous that we have a very expensive mass incarceration system just to enrich lobbyists... rather than for creating a better world and empowering everyone.
We need an e-government infrastructure like Estonia too. It would make so many things like verifications cheaper and more efficient.