I wonder, too....
As said: my thinking only, but it would support the common understanding that the troops in Finland would have been issued Winchesters.
Also it supports your thought about the logistics with which I definitely agree. When looking at service routes (and especially in those days), it would have made sense to have the border troops (let's call them border troops even though there was no border...) issued with the same rifles as the ones in Finland.
Additionally, many of these rifles are encountered in Karelia i.e. Western Russia bordering to Finland which again supports this thinking.

Small country...

The article states (my translation):
"The balance was 220 000 weapons. This included Mosin-Nagants, Berdan rifles, German Mausers, Swedish Mausers, Japanese rifles as well as the Winchesters. In the inventory of 1st Feb 1919 there were 3611 Winchesters. Of these, the military had 1031 and "other authorities" (the police and customs?) had 2580 of the model 1895."
As you can see, no accurate info about the types of troops so supplied. Besides the police and customs the border troops likely will have had Winchesters; the article author also does not know and hence his question mark. I don't know if the border troops at that time were part of the military, although I presume so. Today, the border troops act under the Ministry on Interior, like the police; although the organization is rather military-like, and you can even do your military service in the border troops.
In Finnish history, WWII is divided into various time periods. During the peace period (time period between the Winter War and Continuance War) in June of 1940 the military books held 1697 Winchester rifles.
As can be seen, some weapons had been taken from the "other authorities" in the use of the Army.
The main weapon at that time was the Mosin-Nagant based "Pystykorva", model 1939 if my memory serves me correct; I have never really taken a keen interest in this model, I'm sorry to say.

The quote above also gives you an indication of the various weapons that had found their way to the young nation's arsenal. Mostly, weapons were obtained from Sweden - quite a few of the Swedish Mausers m/96 were shipped here. My brother recently obtained one. I would have posted pics had I not lost all of my photos in a recent computer transition. I'll shoot some more and post.
The long traditions and connections to Germany also ensured a good supply of various other Mauser rifles.
Why the Japanese Mauser? Dunno! Perhaps also these had been issued to the border-land's secondary troops after the Russo-Japanese war?
Lots of weapons were imported from various sources during the 1918 civil war, and then you couldn't bee too picky about what you got. The Soviet Union for sure supplied the "reds" with weapons, and that many of those were Winchesters I think is likely. It would also partly explain the "Vintovka" nomination, i.e. that the Soviet rifles were associated with or as Winchesters.

A big thanks to Lancaster for starting this highly stimulating thread!

- Lars