It's actually pretty hard. The first problem is that if you paid them the amount it would take to attract good people, it's easy for opponents to score cheap political points by charging you with wasting tax dollars or comparing the high salary to that of unskilled workers. Meanwhile a high salary increases the perverse incentive to fill the position with cronies as a reward for political support, which is the opposite of helpful.
There may be ways around it if you're creative, like making a benefit of the position that you and two generations of your descendants get free healthcare and college tuition / loan forgiveness forever (provided you do the job for at least two years). Then that's worth potentially quite a lot of money and provides a suitable incentive to attract good people, but makes it harder to take cheap shots because the total cost is indeterminate in both time and amount. Meanwhile you attract just the sort of experienced greybeard you want, who can do the job for a few years as the final leg of their career before retirement and in so doing provide financial security for their children and grandchildren.
At which point it becomes sort of a meta problem, because to bring that about you need honest and creative people in the legislature to make good rules, which was kind of the original problem -- how do you get good and honest lawmakers and exclude incompetent and corrupt ones, when the people choosing them have insufficient time and expertise?
> Given that regulatory bodies really ought to be non partisan and focused on a particular field or industry of activity, perhaps the institutional heads could be selected via a more democratic process (rather than just whichever buddy helped out an elected representative win their campaign) and given guaranteed minimum terms?
Making them directly elected positions might help, but it's more like trading one set of problems for a different one. Now it's an elected position, but are random airline passengers going to care about who gets elected? (Boeing will, of course.) How do you get honest specialists who know what they're doing to even run, and how does a random voter tell the difference between that person and an industry shill prior to putting them in office? How do you prevent the foreseeable result when there is a disaster not long before an election and people elect a strongman to Do Something About It, only to ruin everything with bluster and incompetence?
> And as to the problem of turnover throughout changes of political administration, isn't that perhaps also merely a policy problem? Why does the FCC or HUD need to be gutted and repopulated during a transition of political power?
It's because they make policy. If a Democrat replaces Trump they're certainly not going to keep his head of the EPA. Giving them guaranteed terms that are of similar length to the President's doesn't strike me as something likely to change much in a helpful way -- and you still have the same issues in choosing who to select to begin with.