It all depends really how you make them. If you for example give them few vehicle and gun units, but make them better than the soviet counterparts (even if it means unrealistic measures), then you would have only couple of vehicles and guns to play around... and as we all know COH, whats the fun in playing with just couple of whatever you can produce?
I would see Continuation War a lot more potential than Winter War because in Continuation War the odds were more balanced, meaning we had something to take out the Red Army tanks. I would imagine the game would work better in a single player campaign with the finns, rather than in a skirmish or multiplay, because in the campaign you could simulate some of the situations of real life battles with scripts. In multiplayer, that is quite hard, unless of course the map design is so nazi it wont allow maneuvers that werent there irl. If the battles would only involve the -44 battles, which are one of my favorite subjects, it would perhaps become quite good gameplay wise, given some unrealistic liberties to the coding of the finnish vehicles.
I personally would imagine the finns being normally equipped, no matter which doctrine, with at least little artillery and tank support, but all doctrines would mostly play out as infantry, never the less.
"Artillery Doctrine" would allow the player to use the ridiculously effective and precise finnish artillery system which at the time was the best artillery system in the world. We perfected the effective use of minimal amount of barrels. This would mean quick offmap artillery in the forms of peite (cover) which moves forward as a creeping barrage, allowing attacking finns move safely behind the barrage. Sectorial defense, one could set a number of locations to be watched over. Lets say the area A and area B. When enemy attacks on area A, all guns could be targeted to only that area, but if the enemy attacked both areas, same function would let all barrels fire on both areas the same time. A lot like the british artillery doctrine, except you would have the artillery offmap, it would be more effective but more expensive to use. Also, there could be a finnish Messerschmitt strafe runs available. Finns would normally also have some in-map artillery units, which could be moved and dug in. But their range would be more limited. This would be the basic artillery unit for all doctrines.
"Kuhlmey Doctrine" would create a mixture of available call-in troops which ultimately ends up to the ability to call in Kuhlmey Stuka's and Fockewulfs to strafe targets, such as those important bridges or enemy artillery. Kuhlmey doctrines first abilities would involve anti-tank infantry units which you could call in from off-map. These would be bigger squads armed with couple of panzerfausts per soldier and even one panzershreck. Armed with Suomi submachineguns and rifles. While the normal finnish squad would carry rifles, they could be upgraded to have Suomi's, DP's and Panzerfausts.
Kuhlmey doctrine would also grand you the ability to use recon runs with the fast fockewulfs.
"Lagus Doctrine" on the other hand focuses to the tank warfare which would be costly to use, but effective when used correctly. Losing these units would make it feel like a knife through heart because they would be so valuable. Lagus infantry ability would let you call in squads of infantry who are specialized in combat and may use the FireUp ability, making them hard to suppress by MG/tank fire. Ferocious fighters. Then next tier ability would be call in tanks, a group of three fast moving BT-42 self-propelled howitzers for the infantry support role (StuH replacement more or less). An ability called Lagus-Group could consist of the legendary Mikkola' Squad which would involve Two T-34/76, two T-28, one KV-1 and one T-50 command tank. Highly effective to counterattacks as they have a mixed group of tanks. The KV-1's ability to absord heavy damage, the swift T-34's to knock out enemy tanks, T-28's for infantry support and T-50 for commander ability such as improved moral to all units effected in the radius of his vicinity , rush-command and binoculars. The ultimate ability on the last tier would of course be the Lagus Sturms. A group of four Stug3G's available to counter the enemy tanks in either fierce counterattacks or simply setting up ambushes on defensive lines. The finnish Sturm would be one of the most feared units of the Red Army players.
Just my wild imaginative 2 cents.