I am actually gonna take the opposite tack here... While I would definitely eliminate the Dept of Transportation IF I Were King For A Day, I would retain the NTSB attached to some other agency.
This incident is a fairly rare case of them going outside their usual boundaries, but the NTSB is the best in the world as what they do: accident cause determination. This government function saves lives, and has for decades now. It's also an operation that is unfeasible to perform at the state level, and impossible commercially.
So I dunno - maybe a government Transportation Safety Agency to hold the Traffic Control, [a reduced] FAA, and the NTSB while scrubbing virtually everything else in the DOT, but this group does need to be preserved. It is normally one of the few government operations that is worth our tax dollars.
That said, they are in the business of issuing reports and making recommendations. Elected officials are then charged with enacting the recommendations (or not), and that's the proper structure, IMHO.
Or, Congress COULD strip all federal agencies and departments from issuing regulations that have the force of law. Congress gave them the authority and Congress can rescind it.
If the regulation must have the force of law, it should have passed both legislative houses and should have been signed by the President.