In as much as my private key gives me full control of my funds and it grants me access to sign a transaction if it gets misplaced then I'm doom for life. What if there was an improvement on this maybe a way of recovering back ones private key if it gets misplaced even if it would take a proof of work alogrithm or even a more tudious process but having a chance to get back lost wallets in cases of carelessness or maybe hazardous damage since it's not advisable to store private keys online nor would it possible for everyone to memorize this combined alpha-numeric values of SHA256 alogrithm
there is no need to change or break bitcoin by making it hackable for forgetful convenience. it just requires people to have prior thought about how they in the human world backup-keep safe their keys in the first place
bitcoin already has
many ways to protect a private key. and many other ways for humans to protect the keys
you can first create a key as a seed of words, easier to memorise..
which you can then personally decide a multitude of ways to preserve it, like engrave it in metal and bury it in your yard.
or turn the bytes into a cypher which you know the puzzle to uncypher and then just have a million copies around your house which are meaningless if seen, because only you know the cypher to translate it
having a system that requires PoW to recover a private key, pretty much means needing PoW to create the key in the first place.
which then becomes a wall. where it starts costing people something to make a key...
people would use bitcoin less if it started costing them something just to have an address/"account" before they are even able to receive funds in it.
after all would you use a paypal account if they asked for a $X fee just to sign up to them?.. nope. thought not.
as for the amount of work needed. well some dormant keys have 50coin+ on them per key (satoshi stash) ($2m+ each)
so if it only cost say $2(reasonable service entry fee) of PoW to recover(gain access to a key) then thats a way to cheaply hack keys that should be unhackable. so ofcourse if there was a system. the 'entry fee' to create a key/recover a key would have to be a few $m atleast
no one would pay $2m to enter a service they never used before(one end of the scale) and no one would want to risk their wealth on a key system that can be hacked for just $2(other end of the scale), so either way.. it just wont have any real utility as a feature without breaking the overall utility of bitcoin