We should, and then they'd probably stop killing tens of thousands of people every year (cf http://www.fdareview.org/05_harm.php and studies cited within)
I haven't looked into it enough to have an opinion one way or another. I suspect that the article is cherry-picking, but perhaps the FDA is so critically fucked up that randos off the street could do a better job. But this isn't about the FDA. You're claiming that knowing a lot about economic regulation makes you a superior domain expert in any field which can be regulated. That economists are better qualified than doctors to regulate dangerous medicines, better qualified than ecologists and meteorologists to predict human-caused climate trends, better qualified than automotive engineers to determine what makes vehicles safe. That's very silly.
"You're claiming that knowing a lot about economic regulation makes you a superior domain expert in any field which can be regulated."
Not quite. I'm claiming that knowing a lot about regulation makes you more qualified to study the effects of various regulations, after listening to input from domain experts in the field.
An automotive engineer can tell you that requiring component X in all vehicles would make accidents Y% less fatal - an economist can tell you whether that actually translates to lives saved (maybe it increases the risk of accidents due to behavioral changes? This was a theory around seatbelts for some time although I can't speak to the current consensus in the field).
Sure, an economist who isn't allowed to speak to the relevant domain experts wouldn't do well. But likewise, a domain expert who doesn't know economics wouldn't do well either. Understanding side effects of regulation, incentives, etc are critically important for crafting regulation/public policy that actually help.
Wait, what? Knowing a lot about regulation in general doesn't mean you know shit about any given subject of regulation in particular.
By your logic we should fire the doctors and scientists at the FDA and replace them with economists.