A 10mm wouldn't leave you much squirrel or bunny for the stew pot though. The question wasn't what is the best hand gun for bear, or target shooting or plinking or small game,or self defense. The question is "what is the best all around handgun?" A .22 kit gun might be great for plinking and putting small game in the pot but won't help you much if you get attacked by a bear. A .480 Ruger would ruin a bears day but would not be a lot of fun to plink with and would vaporize squirrels.
So if I was to take a logical approach to the Op's question the first thing I need to think about is what caliber comes the closest to being adequate (not necessarily the best) for all of the above mentioned purposes. I would have to consider the availability of ammo...Can I buy ammo for it if I find myself in Podung, Maine? Is there a wide assortment of factory loadings or components available for it, can I buy or roll... target loads?, small game loads?, large game loads?, self defense loads?
IMO these criteria narrow the choices down to .357mag/38spl and .44mag/.44spl. I think that the .38s are generally easier to find and come in a wider selection of bullet styles. So I will go with a .38mag/.38spl even though I would rather have a .44mag if I ever had an encounter with a bear.
Now I can choose a platform to deliver these rounds. Well, obviously the .357mag/.38spl are revolver rounds so I can rule out semi-Auto pistols for that reason alone.
Okay now the question of barrel length for my revolver. A 2" snubby is easy to pack but doesn't give me much of a site radius for precision shooting such as needed for target shooting or hitting small game. A 7" or longer barrel is way to unwieldy and heavy for me to carry all the time. IMO, a 3" to 4" barrel is a good compromise.
Single or double action? I would go with a double action. Reloading is quicker and I can still cock the hammer and shoot it SA for better precision if I choose.
Sites. Fixed sites are less delicate, but if I want to use this gun for multiple loads for everything from small game to bear defense so adjustable sites are a must to me.
Okay so now I have a 3"-4" DA revolver with adjustable sites in .357mag/.38spl. What brand do I go with. I want a proven track record, good customer service and parts availability and reasonable cost. S&W and Ruger are the only two I know of that fit the bill and I would be happy with either one of them.
Now, I'm not going to say that my choice is the best choice of an all around hand gun for everyone. You may have a different set of criteria than I. But I just wanted to explain the thought process behind my choice and show what choosing "the best all around handgun" might entail.