As usual in questions of this sort, one wonders "Best? In what regard?" That is the sticky part.
The Japanese rifles suffer from the Japan civilization having no tradition involving firearms; certainly not at the time. For that matter, neither did the Russians. Looking at the Mosin-Nagant Rifle and the Russian/Soviet handguns, one finds them quite similar to the Japanese Type's 38 and 99 and Nambu handguns. Made with an eye toward 'utility' on a superficial basis and only the briefest attempts at finishing. Both nations' weapons are crude looking and rather ugly. Make that really ugly. Neither groups of rifles or handguns had any thought to trigger use, and the sights are about as dismal. However, I would rate the 1891 Russian rifle very high in the "... ran out of ammo and am now clubbing anyone in my reach ..." category. They are sturdy.
Just about all the Mauser rifles (I'm thinking in terms of the South America 7x57mm rifles, the Swedish rifles, the Argentine models and so forth) were gorgeous. Even the WWI produced 1898 was smooth, silky and well finished. The bolts worked well, the rifles fed well, and they shot to much the same place. If one could figure out the barleycorn sights. And the triggers were long and creaky, but could be learned.
Italy used the 1891 Carcano much of WWII. The 6.5x52mm Carcano threw a 160 grain bullet at about 2300 fps. So it was about the same as the 6.5mm Swede ballistically. Any condemnation of power is based on lack of information or a spurious comparison to something else. The trigger wasn't great as issued and quite possibly the training wasn't up to par, either. It shoots okay, but not impressively.
The U. S. rifles had good aperture sights. Trigger pulls were mass produced, but not as bad as they might be. (Not as good as they might be, either.) They were well made, rugged and reliable. They were quite accurate as well.
The French were adaptive and quick to 'evolve' mechanically. The WWI era Lebel and Berthier rifles were more or less slugs, but born of a 'hurry up' and bureaucratic mentality. The M36 MAS was much better in design and manufacture. The MAS would be more admired had the French been able to use them much.
The No 4 (Lee-Enfield) rifle stood up and delivered as needed. It wasn't as accurate as the U. S. rifles in a point target sense, but would usually deliver 'minute of villain' accuracy in combat. Which frankly, is sufficient unto the day. They were reliable and rugged as well. And (foreshadowing the Soviet Union) they were manufactured, along with ammunition all over the Empire. Were I entering an area where I knew I was going to be fighting for my life, I'd opt for a No 4.
For building a sporter (replace the stock, sights, trigger and refinish), I'd likely favor the Mauser. Except for the Mannlicher, which seems to have ended shortly after WWI. Target shooting - the NRA "National Match" course, I'd rather have a M1903A3 Springfield (with match armorer preparation). For marching, which ever one is lightest.
