I want to reply with the complete number of 2 at the power of 256 but that would take quite some forum pages space and how would you possibly steal from such a huge number, the probability to get it right it is negative, less than 0 and that is why the Bitcoin algorithm is one of the strongest. We also have ready made post quantum computers cryptography algorithms and encryption so I wish you luck in your idea and let us know when you will be successful in the year 2140 after all of the Bitcoin has been mined, maybe up until then your theory can hold up and prove itself true  .

I have a number in my head, it is between 1 and 5. I'll give you 5 tries to guess that number. Did you get it right? Cool.
Now I have a number between 1 and 1000000 and I'll give you 5 tries to guess it. Did you get it right? No? huh.
Now I have a number between 1 and 115792089237316195423570985008687907852837564279074904382605163141518161494337 and I'll give you billions of guesses. Did you get it right? No. You can't because you aren't fast enough in guessing numbers before the universe ends. And that's what the bitcoin private key range is.

That's exactly what the puzzle is proving as well. You can spend lots of computing power, even pool the work with others and by aggregating your computing powers be able to go through a much larger number of keys but it still is not enough to break a key that is in a much smaller range, as it is evident from how increasingly difficult it is getting to break the subsequent keys in that puzzle. It is impossible to break the 2²⁵⁶ range.

Because they are in a much smaller range (between 2⁰ and 2¹ then between 2¹ and 2² and so on) which significantly decreases the number of keys to check (as opposed to being between 2⁰ and 2²⁵⁶) making the time short enough to become possible to break.

Do you mean public key? No, hashes are irreversible meaning you can not compute the public key from its hash. Knowing the range of the key does not give any kind of advantage on that either, because the hash is still irreversible.

The biggest, actually.
Even with talks about quantum stuff that would be able to work around such tasks - it will be in the times when there are probably counter-measures to it. And for it to work, it should become mainstreamed for everybody, which is clearly not the reality we are in, it would be hella expensive and not be targeted at BTC first and foremost.

"All BTC in mempool can be stolen"

The only way this statement makes sense to me is if it is perhaps a language barrier thing and OP is talking about double spends.  I don't think that's actually what is happening, but I thought I'd throw it out there.  The only time I've ever seen BTC get "stolen" from the mempool is during periods of expensive fees when someone does a double spend.  I could see someone on the receiving end of the original transaction feeling like their BTC was "stolen" from the mempool in that circumstance...

Actually public key helps moving the balance (BSGS method), which was proved in bitcoin puzzle search. For any random private key with balance this only changes the range which private key is in.