First, thks guy desperately needs PR - his hero self-image makes him look even worse than Shkreli. In this case, all they'd need to say is that Mr Corvin is interested in moving products in areas with low demand, such as rural Tennessee and Kentucky, to people in desperate need in areas with high demand like the big coastal cities or for example those with certain immune disorders. That's all they should say.
Second, let's try to separate feelings for the person (Corvin) from the activity (price gouging). When there's a shortage between quantity demanded and the quantity supplied, there has to be some form of quantity restriction, whether that involves price increases, queueing, quotas or even lotteries. The alternatives are not a person buying sanitizer for $10/bottle on Amazon versus $1/bottle at Safeway, but between $10/bottle on Amazon versus either not getting it, waiting in a long queue at a store (which is counterproductive to social distancing), having some sort of quantity restriction (which is really lousy for people buying for more than just themselves). It's not clear to me at first glance which is worse.
Ultimately, the shortage problem can only be mitigated by additional production or a return to pre-crisis demand. There are probably quite a few producers who could temporarily retool in order to produce sanitizer or other supplies that are in high demand, but this would be costly and risky and disrupt their supply chains and distribution. They could only justify the move with significantly higher prices than the pre-crisis market price. Any such firms would be crazy to do this, given the legal proscription and moral outrage incumbent with price increases during a crisis. There might be a few firms that try to enter as a PR move, and maybe in some places, the government will subsidize production of demanded goods at a low price, but these are not sustainable options. By having a very absolutist stance against price gouging, we are preventing the potentially ameliorating effects of additional production albeit at a higher price.
Third, I am a little disgusted with the NYT for publishing this story. While they can't be blamed for Corvin's diarrhea of the mouth, they did find the person to profile who is maximally unsympathetic person to the NYT's audience: ex-military, lives in the South (and judging by his yard, a hillbilly), not well-educated and has a job (Amazon reseller) that is low status. It may be newsworthy that someone in rural Tennessee has 17k bottles of sanitizer, but I can easily imagine the sort of entrapping questions that the reporter used with this guy, and it makes me, well, irritated. (Anyone who has had experience with a hostile reporter and went in unprepared knows what I'm talking about.) The NYT is taking advantage of this crisis to drive outrage clicks, just as much as this guy is taking advantage to profiteer. In the process, they're making this guy the public face of "price gouging" during COVID-19, despite him being very much a private person unprepared for the scorn about to be unleashed on him and his family.
I think the best way out of this is for someone with charitable intent and the monetary means to buy up his supply close to at-cost (I'd guess $40-50k), and then distribute them to particularly vulnerable people at low prices or even gratis. Corvin himself could maybe do that. If he did have a change of heart, I somehow doubt the NYT would cover it so prominently, if at all. For the sake of all parties, I hope I'm wrong.