I disagree. There are a lot of factors at work here, but some of the reasons this rationale would backfire on Sony are as follows:
1. The Console market does not work like the PC market. Marking a console at a low-end PC price tag would kill that console in the consumer's eyes.
2. Scalpers are not actually making $1000 of PS5s, and going by Ebay most of them are maybe getting up to $800 but not much more. So...I guess an argument could be made for Sony selling them for $800, but see #3.
3. Sony's pricing doesn't exist in a vaccuum. It is in competition both with Microsoft and its own legacy of pricing. They brought out the PS3 15 years ago at $600 and it crippled their potential gains for that entire console generation (and prompted price reductions and slimmer models almost immediately). Consumers will not look favorably on a $1000 PS5 if the Xbox doesn't follow suit....and Microsoft clearly didn't need to; my own household has had no trouble finding three Xboxes (2 Series S and one Series X; don't ask) and as a result Sony would price itself out of the market if it did this.
4. The people willing to pay scalper's prices are a distinctly smaller subset of possible buyers. They are not most people, and I suspect more than a smattering of those sales have come from squandered relief checks.
5. Even if Microsoft and Sony both collectively decided to price fix at $1000 all they would do is start drawing attention to the value of their product against low and mid range gaming rigs, which often perform as well as the current generation of consoles anyway. People will be forced then to consider the ease of access consoles sell against the versatility and freedom in the market that gaming rigs can offer, and that is not a comparison that Sony or Microsoft really want (well, MS might not care as much....)


 

It might be economically sensible, but an awful lot of potential future customers would hold it against them.
 

And its quite possible that if they charge more for their console, the extra cash will end in the shareholders hands and not exactly at the factory.
 

This is a question that has occurred to me, too. If scalpers can make money by reselling consoles, then Sony is clearly selling their console at below market price. But as Doctor Futurity says, there may be long-term PR reasons why they have to do that. If they launched the console at, say £750, with the sort of supply they did, they would clearly need to drop the price once the high-spenders had got the console and supply increased, or they'd lose out in the long run.

Don't rule out the possibility of simple mistake, of course. It's quite possible the scarcity of the console wasn't entirely planned.