Of course Sony would have to design the chip with the specific target of it being able to play PS4 games natively, so zero resources would be required to play that library on the handheld. It's not analogous to porting PS3/360 games to switch, just like PS4 games don't require porting to PS5...
Again, maybe this ain't possible, but we're talking about matching very antiquated hardware now in terms of PS4.
The steam deck is doing something similar in the PC space, and it's hardware would be easily eclipsed by a PS equivalent.
Re. your last point about PS5 games, Ratchet & Clank was supposedly only possible via the power of the PS5 hardware, and what do you know, steam deck offers a very playable experience, and again a PS equivalent would be far more capable.
We'll see a lot of 3rd party games ported/playable to/on switch 2/steam deck (and it's likely successor), my understanding is that unreal engine 5 has a lot of tools geared towards scalability.
It's not as easy as "just design a chip to play PS4 games natively" though, portable hardware is necessarily different from console hardware, so games built for PS4 would need to be reworked to run well on it.
As for Ratchet on Steam Deck, that did require work; the ability to run at lower specs was a part of porting it to PC.
There will be PS5 games ported to run on Switch 2 and Steam Deck, just as there are PS4 games ported to the current Switch, but none of that comes for free, it all ties up development resources.
A new handheld would require a big investment and pull vital resources away from the PS5; they tried this before with the Vita and the result was disastrous.
I'm not suggesting it'd be easy, like any chip it'd require up-front investment. Numerous consoles have had backwards compatibility, though admittedly you'd have an added layer of difficulty working with a mobile SOC.
What I'm suggesting wouldn't require any work on the pre-existing PS4 library though, they'd run natively (or I suppose, be emulated at such high efficiency that it could work on a modern mobile chip).
I genuinely don't know if it's possible, if you are an expert & know for sure it's not doable with today's tech then I'll take your word for it.
Also I've not claimed that porting a game is free, but it's orders of magnitude cheaper than creating games from the ground up for a single platform.
Developers would do their maths & decide whether the juice would be worth the squeeze, and that itself would come down to how well the platform is doing. As with Ratchet, games will be getting scaled down to run on low end machines, including a considerably weaker steam deck, so much of the work will already be done.
A PS Deck wouldn't require any platform specific software so wouldn't demand resources like the PSP or Vita.