Going back to OPs original proposal, or my understanding of it:
Idea: design a mining algorithm that requires a special CPU (≈ASIC) that runs at a low frequency, but gets a shit-ton of specialized work done per cycle (≈ASIC).
Outcome: OP suggests that such a chip (≈ASIC) would be more expensive to manufacture (≈ASIC), but require less electricity thereafter, thus be more efficient (≈ASIC).
So how does PoW work and why will your chip not help?
The way PoW works is that miners have to expend valuable “real world” resources to validate the network and then get rewarded with coins in return.
It doesn’t matter if it’s electricity or capex or whatever that is being expended, it just has to be considered “valuable” by a large enough fraction of the population. Given the world is still full of boomers, Satoshi has chosen electricity as the valuable resource. Every boomer understands that building a power plant costs a fuck-ton. Besides, transporting energy is relatively straight forward (over short-medium distances) and energy is widely available.
If the mining mechanism is changed, then the total expense has to be maintained, or network security suffers because it would become cheaper to attack the network.
If the electricity need per miner goes down, then there will have to be more miners to maintain security. Alternatively, some other resource will have to be expended more, like the specialized chips that you describe . This is unlikely to be any better for the environment than today’s PoW, I’d rather expect the opposite. (See ASIC waste discussion.)
How is PoS different from PoW, and is it more efficient?
In PoS, the idea is that you make a big upfront investment representing your primary cost. After that, maintenance cost is almost negligible.
However, you don’t invest in something that boomers understand (such as hardware and energy), you instead invest in some magic Internet money coin (ETH) that is not a traditional “real world” resource, yet.
There are at least two criticisms with this model:
Network security depends on a virtual resource (ETH) that a big part of the population (e.g., boomers) does not understand and value appropriately. Everyone sees the intrinsic value of energy and hardware, but many people don’t see the intrinsic value of ETH. If people don’t think that ETH is valuable, then ETH can’t protect a blockchain to the same extent that PoW can.
Requiring a big upfront investment into ETH, with negligible cost afterwards, and that entitles the owner to an infinite stream of future coin supply, will over the long run (several human generations…) create huge inequalities. To restore equal opportunities, it will require massive taxation just like in the current boomer system. A model that requires constant expenditure of real world resources doesn’t have this issue.
Personally I think that as of today, PoS can only be secure as an L2 on top of a strong PoW chain. This might change over time, as a larger fraction of the population starts understanding the intrinsic value of virtual stuff. The fact that NFTs suddenly entered mainstream art circles is an encouraging sign and gives me hope that PoS can at some point replace PoW. But I don’t think now is that time.
 While not literally getting expended, the chips become useless quickly given the technological progress.