Sorry but no. No, they didn’t want to but couldn’t. The Finnish troops were stopped and there was a stalemate when both sides did not have the strength to solve the problem.
In August 1941 the Finns reached their pre-1940 border on the Karelian Isthmus and in some places crossed it to straighten the frontline (the old border formed a wedge into the Finnish side of the Isthmus), this necessitated punching through the Russian Karelian Fortified Region (KaUR) in some places which was done. The divisions expected to continue the advance, and plans for continuing the advance on the eastern part of the Isthmus were drawn up, but Mannerheim who was not interested in the destruction of St. Petersburg (Leningrad), deemed sufficient defensive positions had been reached and halted the advance.
The Finnish divisions remained in good condition and the strength of the Finnish forces near the KaUR on the Karelian Isthmus in September 1941 was 2.5x that of the Russian defenders. Strength figures from 6.9.1941, picture is self-explanatory and needn't translation:
Excluding the 3 artillery and 1 heavy artillery battalion of the 8th Division and the coastal artillery reserved for coastal brigades still in the process of forming, the Finns in the beginning of September 1941 had 184 light cannon, 44 medium howitzers, 40 medium cannon, 68 heavy howitzers and 18 very heavy howitzers near the vicinity of the KaUR. The 8th Division was further back on the Isthmus near Koivisto, but could have been used in puncturing through the KaUR if so desired. Similarly, coastal artillery could have supported the advance as it had done during the reconquest of the Karelian Isthmus in the past month.
Table of Finnish artillery formations and equipment on the Isthmus at the beginning of September, excluding coastal artillery:
Guide to units:
I/KTR 11 = First battalion of Field Artillery Regiment 11
Rask.Psto 21 = Heavy Artillery Battalion 21
10. Jär.Ptri = 10th Very Heavy Artillery Company
Patteristo, shortened psto. = battalion-sized artillery unit
Patteri, shortened ptri. = company-sized artillery unit
Guide to equipment:
9/203 H 17 = nine 203mm model 1917 howitzers
12/105 K 29 = twelve 105mm model 1929 cannon
H = Howitzer
K = Cannon
The Finns facing the KaUR enjoyed approximately a 2:1 advantage in the amount of artillery compared to the Russians, and over 2:1 in terms of firepower, taking into account the more numerical heavy artillery the Finns possessed. The Russian 23rd Army on the KaUR had under 200 guns, of which only 22 were 152mm.
The reconquest of Finnish Karelia had proven the Finns were fully capable of breaching heavily fortified defences, and it is likely the Finns could have punched through the rest of the KaUR and cut Leningrad from its access to Lake Ladoga, resulting in the city's capitulation some weeks later. The strong rear fortifications of the KaUR known from later during the war did not yet exist in September 1941.
The above pictures and most of the information is from a thesis called
The actions of the Finnish Army against the Karelian Fortified Region in September 1941 (
Suomen Armeijan toiminta Karjalan linnoitusaluetta vastaan syyskuussa 1941).
Similarly, and one could say even more certainly, the Finnish Army was completely able to cut the Murmansk railway by spring 1942 with a strike to Sorokka (Soroka/Belomorsk) from the Rukajärvi-Maaselkä (Rugozero-Maselgskaya) direction. The Russian forces facing the Finns there were weak and withered down from poor supply, and a Finnish detachment that had previously blown up a section of the Murmansk railway in January 1942 had proven how vulnerable the railway was. With the commitment of only a few divisions, the Finns would have outnumbered the Russian defenders 3:1 and been able to reach Sorokka and cut the Murmansk railway, after which it was planned that defence of the Sorokka area be handed over to the Germans.
Here I'm using another thesis, called
The Sorokka-plan: The importance of information from reconnaissance in the preparation of the operation during January-March 1942 (quite a mouthful when translated to English from the original:
Sorokka-suunnitelma: Tiedustelutietojen merkitys operaation valmistelussa tammi-maaliskuussa 1942), which goes to great lengths of assessing the feasibility of the plan and concluding that it would have been completely possible to carry it out successfully with a few divisions.
Mannerheim stopped the Finnish Army, not the Russians. The reasons for this were mainly political: the Finns had achieved their goals of reaching defensively favourable terrain, wanted to maintain that they were in a separate defensive war against the Russians and not a part of the German war of conquest, and did not want the Americans to declare war on them, which they would have, had the Finns permanently cut the Murmansk railway. And then there were Mannerheim's personal feelings on the city of St. Petersburg, having spent much of his life there, which I think were the primary reasons behind his reluctance to engage in any kind of hostility towards the city.
Also if you want to continue this, I suggest it be done in a new thread.
EDIT (to avoid double-posting):
Just to clarify, what I meant was that Finland doesn't really need neither a focus tree nor historical flavour (generals, graphics, companies etc) to make the game live up to a decent historical standard.
You can't reach historical performance as Finland with historical force ratios, not even close. You'll be curb stomped even by AI SOV under those conditions.
See e.g.:
Ironically, 13.3.1940 the Russians had 58 divisions on the Finnish front, another 4 nominally under the Finnish Democratic Republic, though these 4 were less than half the size of a Russian division, and another 10 divisions in reserve. Furthermore there were 3 airlanding brigades, 1 infantry-MG brigade, 8 tank brigades, 10 tank regiments, various border and reinforcement brigades. These amounted to 11,266 artillery guns and mortars, 2,998 tanks, 3,885 aircraft, in total about 1,000,000 men.
Those figures are off, for both sides. Considering that these figures are supposed to include all forces that participated in the war, even if it's not simultaneously, a more accurate listing of statistics would be:
Soviet Union
1,500,000 men (200,000-300,000 KIA/MIA, 300,000-600,000 WIA, 5,000 POW)
c. 6,541 tanks (3,179 were "lost" on the Karelian Isthmus alone, hundreds in other sectors, of which 2,200-2,500 to enemy action, the rest to breakdowns. Hundreds repaired or recaptured post-war)
c. 4,500-5,000 aircraft (1,000 lost, possibly more, c. 650-700 to enemy action)
Finland
372,000 men (25,904 KIA/MIA, 43,557 WIA, 1,000 POW)
c. 13+ tanks (13 Vickers 6-tons, a handful of WW1 Renault FT-17s that were only used as static, dug in defences. 8 Vickers 6-tons lost in combat/due to breakdowns. A number of Renault FT-17s were either lost or abandoned where they were after the peace - recaptured in the Continuation War as they were so obsolete even the Russians didn't bother taking them)
c. 230 aircraft (64 lost, 47 to enemy action; 35 to enemy fighters, 8 to AA)
Navies also played a role, e.g. the Red Banner Baltic Fleet had orders to cut Finland's vital trade routes, but failed in this. I won't go into detail regarding navies though since they weren't mentioned and weren't ultimately
that relevant because of the ice. And honestly because writing this post and checking all the figures has taken too long already.

This Finnish Wikipedia article covers naval activities during the war in some detail, use Google Translate if you're interested (the English article on the topic is lackluster by comparison):
https://fi.wikipedia.org/wiki/Talvisodan_merisotatoimet
Minor corrections to the above: there were 1.2M Red Army soldiers on the front 13.3.1940, the total amount of Finnish participants in the WW number at around 400k, and there were about 38 Finnish tanks (20 Renault FT-17s and 18 Vickers 6-tons). Also, taking into account the casualties the Russians received, and their frontline strength at the end of the war, the number of Red Army soldiers that participated in the WW is likely more than 1.5M, probably closer to 2M.