Multiple times in the later books, Dumbledore points out that he is not perfect. Especially when it comes to artifacts of great power and temptation. It's possible that Dumbledore was afraid he might abuse the power of the stone - or for that matter, simply take it out to look at it at the wrong time, and thus allow it to fall into Voldemort's grasp. The way he did hide it required a good, selfless person to retrieve it and so was very nearly a perfect defense; perhaps he couldn't even have drawn it out himself.
I think the question asker might be putting too much stock in Hagrid's assessment of the pouch, as well. Other things that were more fallible than he claimed include Gringotts, Fluffy, Hogwarts Castle, Hogwarts faculty, Grawp, and Dumbledore himself.
As mokeskin can clearly be cut, because mokes can be skinned, there exist multiple magical and mundane ways of retrieving items from them. Moreover, as Fantastic Beasts details the exact way the pouch functions (only when physically approached by a stranger), a Summoning Charm should also work. These pouches are useful, but not infallible. By the end of Philosopher's Stone, indeed, I don't think Quirell would be considered a stranger by the pouch. After all, if the people you see on a daily basis triggered the pouch to contract and hide, Harry wouldn't be able to examine its contents with Ron and Hermione in the tent in Hallows.